A STUDY ON THE EXPRESSIVE POWER OF

BORDEAUX UNIVERSITY 1
Laboratoire Bordelais de Recherche en Informatique
UNIVERSITY OF LAUSANNE
Faculty of Business and Economics
A STUDY ON THE EXPRESSIVE POWER
OF SOME FRAGMENTS OF THE MODAL
µ-CALCULUS
Alessandro Facchini
PhD Dissertation
II
Contents
Abstract (in French)
3
Introduction
Outline of the dissertation . . . . . . . . . . . . . . . . . . . . . . . . .
Related works . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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1 Logics, Games and Automata
1.1 Trees and transition systems . . . . . .
1.1.1 Trees and their undressing . . . .
1.1.2 Transition systems . . . . . . . .
1.2 Topology . . . . . . . . . . . . . . . . .
1.2.1 The Borel sets . . . . . . . . . .
1.2.2 The Wadge game . . . . . . . . .
1.3 Monadic second order logics . . . . . . .
1.3.1 Full MSO . . . . . . . . . . . . .
1.3.2 Weak MSO . . . . . . . . . . . .
1.4 The modal µ-calculus . . . . . . . . . .
1.4.1 Syntax . . . . . . . . . . . . . . .
1.4.2 Semantics . . . . . . . . . . . . .
1.4.3 Another formalism for the binary
1.4.4 MSO vs µ-calculus . . . . . . . .
1.5 The fixpoint alternation hierarchy . . .
1.6 Parity games . . . . . . . . . . . . . . .
1.7 Evaluation games . . . . . . . . . . . . .
1.8 Automata for the modal µ-calculus . . .
1.8.1 Automata on binary trees . . . .
1.8.2 Automata on transition systems
1.9 The Mostowski-Rabin index hierarchy .
1.10 Summarizing remarks . . . . . . . . . .
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I
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The µ-Calculus on Restricted Classes of Models
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2 The Fixpoint Hierarchy on Reflexive, Transitive, and TransitiveSymetric Models
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2.1 Preliminary remarks . . . . . . . . . . . . . . . . . . . . . . . . . 51
2.2 Finite model theorems . . . . . . . . . . . . . . . . . . . . . . . . 52
III
IV
CONTENTS
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4 Characterizing the Modal Fragment on Transitive Models
4.1 Preliminary remarks . . . . . . . . . . . . . . . . . . . . . . .
4.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.1 The beauty of forests . . . . . . . . . . . . . . . . . . .
4.2.2 More on monadic second order logics . . . . . . . . . .
4.2.3 The logic EF and EF-bisimulation . . . . . . . . . . .
4.3 The complexity of conciliatory tree languages . . . . . . . . .
4.3.1 The set of well-founded trees is not Borel . . . . . . .
4.4 Characterizations of EF over infinite trees . . . . . . . . . . .
4.5 Modal=Borel, on transitive models . . . . . . . . . . . . . . .
4.6 Summarizing remarks . . . . . . . . . . . . . . . . . . . . . .
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2.5
2.6
2.2.1 Finite model theorem for reflexive transition systems .
2.2.2 Finite model theorem for transitive transition systems
The transitive and symmetric case . . . . . . . . . . . . . . .
The transitive case . . . . . . . . . . . . . . . . . . . . . . . .
2.4.1 Some technical preliminaries . . . . . . . . . . . . . .
2.4.2 A first reduction . . . . . . . . . . . . . . . . . . . . .
2.4.3 Normalizing the winning strategies . . . . . . . . . . .
2.4.4 Encoding normalized winning strategies . . . . . . . .
2.4.5 The collapse over transitive models . . . . . . . . . . .
The reflexive case . . . . . . . . . . . . . . . . . . . . . . . . .
Summarizing remarks . . . . . . . . . . . . . . . . . . . . . .
3 The µ-Calculus vs the Gödel-Löb Logic
3.1 Preliminary remarks . . . . . . . . . . . . . . . .
3.2 Gödel-Löb Logic GL . . . . . . . . . . . . . . . .
3.2.1 Syntax and Semantics . . . . . . . . . . .
3.2.2 Embedding GL into the modal µ-calculus
3.3 The modal µ-calculus over GL . . . . . . . . . . .
3.4 The modal µ∼ -calculus . . . . . . . . . . . . . . .
3.4.1 Basic notions and results . . . . . . . . .
3.4.2 The unicity of fixpoints . . . . . . . . . .
3.4.3 Collapsing the modal µ∼ -calculus . . . . .
3.5 Summarizing remarks . . . . . . . . . . . . . . .
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II Hierarchical Questions for Tree Languages Definable
Without Alternation
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5 Preliminaries
5.1 Topological hierarchies . . . . . . . . . . . . . . . . . . . . .
5.1.1 The topological complexity of regular tree languages
5.1.2 The Wadge hierarchy of Borel sets of finite rank . .
5.2 Weak alternating tree automata . . . . . . . . . . . . . . . .
5.2.1 Paths and loops . . . . . . . . . . . . . . . . . . . .
5.2.2 Weak index vs Borel rank . . . . . . . . . . . . . . .
5.2.3 The Wadge hierarchy of weak alternating automata
5.3 Summarizing remarks . . . . . . . . . . . . . . . . . . . . .
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V
CONTENTS
6 Decidable Hierarchies for Linear Game Automata
6.1 Preliminary remarks . . . . . . . . . . . . . . . . . .
6.2 Linear game automata . . . . . . . . . . . . . . . . .
6.2.1 A normal form . . . . . . . . . . . . . . . . .
6.3 Deciding the Borel hierarchy . . . . . . . . . . . . .
6.3.1 Patterns menagerie . . . . . . . . . . . . . . .
6.3.2 Effective characterization . . . . . . . . . . .
6.4 Weak index of LGA-recognizable sets . . . . . . . . .
6.5 The Wadge hierarchy of linear game automata . . .
6.5.1 The difference hierarchy . . . . . . . . . . . .
6.5.2 Bestiarum vocabulum . . . . . . . . . . . . .
6.5.3 Computing Wadge degrees . . . . . . . . . .
6.6 Summarizing remarks . . . . . . . . . . . . . . . . .
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Conclusion
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Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
A Computations of Chapter 6
A.1 Basic properties of operations
A.2 Closure by ∨ and → . . . . .
A.3 Closure by ⋄ . . . . . . . . . .
A.4 Closure by sup+ and sup− . .
Bibliography
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VI
CONTENTS
Thesis Committee
• Jacques DUPARC, Professor at the University of Lausanne, Co-supervisor.
• Benoı̂t GARBINATO, Professor at the University of Lausanne, Internal
member.
• Erich GRÄDEL, Professor at the RWTH Aachen University, External
member.
• Gerhard JÄGER, Professor at the University of Bern, External member.
• Damian NIWINSKI, Professor at the University of Warsaw, External
member and referee.
• Johan VAN BENTHEM, Professor at the University of Amsterdam and
at the Stanford University, External member and referee.
• Igor WALUKIEWICZ, DR CNRS at the University of Bordeaux I, Cosupervisor.
• Marc ZEITOUN, Professor at the University of Bordeaux I, External
member.
1
2
CONTENTS
Résumé
Le µ-calcul est une extension de la logique modale par des opérateurs de point
fixe. Dans ce travail nous étudions la complexité de certains fragments de
cette logique selon deux points de vue, différents mais étroitement liés: l’un
syntaxique (ou combinatoire) et l’autre topologique. Du point de vue syntaxique, les propriétés définissables dans ce formalisme sont classifiées selon
la complexité combinatoire des formules de cette logique, c’est-à-dire selon le
nombre d’alternances des opérateurs de point fixe. Comparer deux ensembles
de modèles revient ainsi à comparer la complexité syntaxique des formules associées. Du point de vue topologique, les propriétés définissables dans cette
logique sont comparées à l’aide de réductions continues ou selon leurs positions
dans la hiérarchie de Borel ou dans celle projective.
Dans la première partie de ce travail nous adoptons le point de vue syntaxique afin d’étudier le comportement du µ-calcul sur des classes restreintes de
modèles. En particulier nous montrons que:
(1) sur la classe des modèles symétriques et transitifs le µ-calcul est aussi
expressif que la logique modale;
(2) sur la classe des modèles transitifs, toute propriété définissable par une
formule du µ-calcul est définissable par une formule sans alternance de
points fixes,
(3) sur la classe des modèles réflexifs, il y a pour tout n une propriété qui
ne peut être définie que par une formule du µ-calcul ayant au moins n
alternances de points fixes,
(4) sur la classe des modèles bien fondés et transitifs le µ-calcul est aussi
expressif que la logique modale.
Le fait que le µ-calcul soit aussi expressif que la logique modale sur la classe
des modèles bien fondés et transitifs est bien connu. Ce résultat est en effet la conséquence d’un théorème de point fixe prouvé indépendamment par
De Jongh et Sambin au milieu des années 70. La preuve que nous donnons
de l’effondrement de l’expressivité du µ-calcul sur cette classe de modèles est
néanmoins indépendante de ce résultat. Par la suite, nous étendons le langage
du µ-calcul en permettant aux opérateurs de point fixe de lier des occurrences
négatives de variables libres. En montrant alors que ce formalisme est aussi expressif que le fragment modal, nous sommes en mesure de fournir une nouvelle
preuve du théorème d’unicité des point fixes de Bernardi, De Jongh et Sambin
et une preuve constructive du théorème d’existence de De Jongh et Sambin.
3
4
RÉSUMÉ
Pour ce qui concerne les modèles transitifs, du point de vue topologique
cette fois, nous prouvons que la logique modale correspond au fragment borélien
du µ-calcul sur cette classe des systèmes de transition. Autrement dit, nous
vérifions que toute propriété définissable des modèles transitifs qui, du point de
vue topologique, est une propriété borélienne, est nécessairement une propriéte
modale, et inversement. Cette caractérisation du fragment modal découle du
fait que nous sommes en mesure de montrer que, modulo EF-bisimulation, un
ensemble d’arbres est définissable dans la logique temporelle EF si et seulement il
est borélien. Puisqu’il est possible de montrer que ces deux propriétés coı̈ncident
avec une caractérisation effective de la définissabilité dans la logique EF dans le
cas des arbres à branchement fini donnée par Bojanczyk et Idziaszek [24], nous
obtenons comme corollaire leur décidabilité.
Dans une deuxième partie, nous étudions la complexité topologique d’un
sous-fragment du fragment sans alternance de points fixes du µ-calcul. Nous
montrons qu’un ensemble d’arbres est définissable par une formule de ce fragment ayant au moins n alternances si et seulement si cette propriété se trouve au
moins au n-ième niveau de la hiérarchie de Borel. Autrement dit, nous vérifions
que pour ce fragment du µ-calcul, les points de vue topologique et combinatoire coı̈ncident. De plus, nous décrivons une procédure effective capable de
calculer pour toute proprı́eté définissable dans ce langage sa position dans la
hiérarchie de Borel, et donc le nombre d’alternances de points fixes nécessaires
à la définir. Nous nous intéressons ensuite à la classification des ensembles
d’arbres par réduction continue, et donnons une description effective de l’ordre
de Wadge de la classe des ensembles d’arbres définissables dans le formalisme
considéré. En particulier, la hiérarchie que nous obtenons a une hauteur (ω ω )ω .
Nous complétons ces résultats en décrivant un algorithme permettant de calculer
la position dans cette hierarchie de toute propriété définissable.
Introduction
In this dissertation we study the complexity of some fragments of the modal
µ-calculus from two different but closely related points of view: the syntactical
(or combinatorial) and the topological. The syntactical perspective means that
properties definable in the modal µ-calculus are classified according to the combinatorial complexity of the defining formulae, that is to say according to the
number of alternations of fixpoints. The comparison of two sets of models therefore comes to comparing the syntactical complexity of the associated formulae.
On the other hand, from the topological perspective, properties definable in the
modal µ-calculus are either compared through continuous reductions or by their
positions in the Borel and projective hierarchies, such positions being based on
how many times the operations of projection, countable unions and complementation must be used, starting from simple (open) sets, in order to obtain the
considered collections. To get a first glimpse of the issues to be discussed, we
shall introduce the main object of our investigations: the modal µ-calculus.
The modal µ-calculus is an extension of modal logic with least and greatest fixpoint operators. The term “µ-calculus” and the idea of extending modal
logic with fixpoints appeared for the first time in the paper of Scott and De
Bakker [110] and was further developed by other authors. Nowadays, the term
“modal µ-calculus” stands for the formal system introduced by Kozen [73]. It is
a powerful logic of programs subsuming (almost) all the most studied dynamic
and temporal logics, like PDL, LTL, CTL and CTL∗ , and corresponds exactly
to the bisimulation invariant fragment of monadic second order logic. Therefore these are very good reasons to consider the µ-calculus as the right meta
formal system for reasoning about assertions concerning temporal properties of
dynamic (reactive and parallel) systems with potentially infinite behavior.
What makes the modal µ-calculus very powerful in terms of expressivity is
not simply the use of greatest and least fixpoint operators, but rather the possibility of nesting (alternate) them. This observation naturally leads to the introduction of a measure of this syntactical alternation, generating what is called
the fixpoint alternation hierarchy. This hierarchy, which shape is represented in
figure 1, attempts to capture the complexity of a certain class of models depending on the combinatorial complexity of the defining formulae. More precisely,
µT
the fixpoint hierarchy consists in the collection {ΣµT
n : n ∈ N} ∪ {Πn : n ∈ N}.
µT
The sigma class Σn denotes the collection of models of formulae with at most
n alternations of fixpoint operators, starting with a least fixpoint, whereas the
dual pi class ΠµT
n denotes the collection of models of formulae with at most n
alternations of fixpoint operators, but starting with a greatest fixpoint. When
n = 0, both the corresponding pi and sigma classes coincide with the collection
of models of modal formulae. The delta (ambiguous) classes are given by the
5
6
INTRODUCTION
intersection of the two dual classes immediately above in the hierarchy.
ΣµT
1
>
>
>
>
>
∆µT
1
∆µT
2
>
>
>
ΠµT
0
···
>
>
||
ΣµT
···
2
>
ΣµT
0
ΠµT
1
>
ΠµT
···
2
Figure 1: The fixpoint alternation hierarchy.
From this point of view, understanding the complexity of the modal µcalculus is like having a complete description of this hierarchy, i.e. being able
to answer the following questions:
• can we always find properties that require more and more complex formulae to be expressed (strictness of the hierarchy) ?
• can the position in the hierarchy of a definable language always be decided?
• and, eventually, can we give a (possibly effective) characterization of its
levels?
For the general case, since the work of Bradfield and others [32, 33, 34, 7, 80], we
know that the hierarchy is strict. However, at the moment, we can only decide
the low levels of the hierarchy illustrated above [78, 129, 103]. Concerning the
characterization of its levels, we know for example that the intersection of the
two incomparable classes of the first level corresponds to the class of models of
purely modal formulae, while the intersection of the two incomparable classes
of the second level is the class of models of formulae without any alternation
[77, 103].
The objective of the first part of this thesis is to describe the fixpoint alternation hierarchy when only models that satisfy a certain well-defined property
are considered, such as, for instance, that of being a transitive model.
The modal µ-calculus is further strongly connected with automata theory,
another important area of computer science. Automata are abstract models of
machines that perform computations on an input by moving through a serie
of states or configurations. Through automata, computer scientists are able
to understand how machines compute functions and solve problems and, more
importantly, what it means for a function to be defined as computable or for a
question to be described as decidable. Initially, the considered input was just a
finite string. But over the last few decades, several new types of automata have
been introduced that extend finite automata in various directions. A natural
extension is to consider infinite computations as a model for non-terminating
reactive systems.
7
The connection between automata and logic was first established in the early
1960s in the work of Büchi and of Elgot, who showed that there is an effective
correspondence between finite automata and monadic second-order logic interpreted over finite words. Later, in the work of Büchi, McNaughton and Rabin,
such equivalence was also shown between automata and monadic second order
logic over infinite words and trees. Moreover, thanks to this equivalence, Büchi
and Rabin were able to prove that the monadic second order theory S1S, resp.
S2S, of one, resp. two, successor functions were decidable.
Since those seminal works, the idea of translating a logic into appropriate
models of finite-state automata on infinite words or infinite trees has become a
central paradigm in the theory of system verification (cf. [133] for a nice recent
overview). Through this translation, the model-checking problem is reduced
to the non emptyness problem for automata. This is for instance the case for
the modal µ-calculus, whose counterpart in terms of automata is given by alternating automata with the parity condition [53]. Roughly, alternating parity
automata are non-deterministic automata whose set of states is partitioned between the set of existential states and the set of universal states and such that a
natural number is associated to every state. The computation of an alternating
parity automaton on a infinite words or trees is then given in terms of a twoplayers infinite game with perfect information, called a parity game. A player
is in charge of the existential nodes, while the other is in charge of the universal
nodes. The game proceeds in rounds. At the beginning of every round there is
a copy of the alternating automaton in its own initial state in a certain node
of the tree. During a round, the player in charge of the initial state sends a
copy of the automaton to a successor in the tree and changes the initial state,
all this done according to the transition function. Since a natural number was
assigned to every state, at the end of a play, an infinite sequence of natural
numbers is produced. A play is thence said to be won by the player in charge
of the existential nodes if the maximum number occurring infinitely often in
the sequence is even. An infinite word or infinite tree is said to be accepted by
an alternating tree automaton if the existential player has a winning strategy
in the corresponding parity game. Because we are able to effectively find an
equivalent alternating parity automaton running on infinite words or trees for
every µ-formula, we can thus successfully base model-checking algorithms for
the µ-calculus on tools provided by that class of finite-state automata.
When considering recognizable sets of infinite words or trees, three hierarchies are classically used to measure their complexity: the Mostowski-Rabin
index hierarchy, or the Borel/projective hierarchy and the Wadge hierarchy,
which is an enormous refinement of the Borel/projective hierarchy. Since it
reflects the depth of nesting of positive and negative conditions, the first hierarchy determines the combinatorial complexity of the recognizing automaton
and is closely related to the previous fixpoint alternation hierarchy of the modal
µ-calculus. The second and third hierarchies, on the other hand, capture the
topological complexity of languages accepted by such devices. Indeed, from a
topological point of view, an infinite tree is very similar to an infinite word,
which is also very close to a real number. The Wadge hierarchy in particular enables a precise comparison of different models of computation. Consider
for instance deterministic and weak alternating automata on trees. Knowing
that there are deterministic languages that are not weakly recognizable and
vice versa, and that therefore we cannot compare them by inclusion, how then
8
INTRODUCTION
can we decide what is the more powerful model of computation? The Wadge
hierarchy, defined by the preorder induced on languages by simple (continuous)
reductions, makes it possible. The sole heights (huge ordinals) of the Wadge
hierarchy restricted to the classes under comparison provide more information
then other logical techniques. It is for instance known that the Wadge hierarchy
of deterministic tree languages has height ω ω·3 + 3, and ω ω·3 + 2 when restricted
to Borel sets [92], whereas the height of the Wadge hierarchy of weakly recognizable tree languages, which are all Borel, is at least ε0 [51]. But using the
Wadge hierarchy enables also to compare models of computation on trees with
models of computation on words. As another example, the Wadge hierarchies
for the two previous models of computation on trees can be compared with
the Wadge hierarchy for deterministic context-free word languages [49] or the
Wadge hierarchy for word languages recognized by deterministic Turing machines [55], whose heights are respectively (ω ω )ω and (ω1CK )ω , where ω1CK is the
first non recursive ordinal, known also as the Church-Kleene ordinal. Moreover,
evidences indicate that the two kinds of hierarchy (combinatorial vs topological)
are closely related. Consider for instance once more the case of deterministic
recognizable tree languages. On the one hand, Niwinski and Walukiewicz [101]
have shown that a tree language recognized by a deterministic parity automaton
is either hard for the co-Büchi level or is on a very low level in the hierarchy
of weak alternating automata, and that this property has a precise topological
counterpart. On the other hand, relying on the previous work, Murlak [93] was
able to verify that for this class of languages the weak index and the Borel rank
coincide and conjectured that this coincidence still holds for weakly recognizable
tree languages.
In the second part of the dissertation we study the topological complexity
of a subclass of weak alternating automata, or equivalently of a fragment of the
alternation free fragment of the modal µ-calculus. In particular, this is done by
describing the corresponding topological hierarchies and by verifying whether or
not the correspondence between the combinatorial and topological complexities,
known to hold for deterministic recognizable tree languages, extends to this class
of automata.
To conclude, there seems to be a whole world of subtle and rich connections
between (fixpoint) logics, (parity) automata, topology and games, as depicted
in the next figure:
Model-Checking
Logic of Programs ................................. Automata Theory
..
..
..
..
..
..
..
..
..
..
..
..
.. Acceptance Condition
Complexity of Models .
µ − calculus
..
..
..
..
..
..
..
..
..
..
.
(Descr.) Set Theory .................................... Game Theory
Gale-Stewart Games
Hence, the study of the structure of the combinatorial and topological hierarchies of fragments of the modal µ-calculus, as well as that of their relationships,
9
leads us to a rich web of connections between different fields, thus revealing the
whole beauty and complexity of the theory underlying this logic. We hope that
this dissertation contributes to this enterprise.
Outline of the dissertation
This dissertation consists of two parts preceeded by an introductory chapter,
the first part (mainly) devoted to the syntactical approach, the other to the
topological.
In Chapter 1 we introduce all the relevant fundamental concepts and results
that will be used in the two parts of the dissertation. In doing so, we always
try to motivate and situate the introduced objects and results.
Chapters 2-4 constitute the First Part of our work. In Chapter 2 we
discuss the strictness of the fixpoint hierarchy on three restricted classes of
models of the modal µ-calculus: symmetric and transitive models, transitive
models, and finally reflexive models. We prove that:
(1) over symmetric and transitive models the modal µ-calculus collapses into
its modal fragment,
(2) over transitive transition systems the modal µ-calculus collapses into its
alternation free fragment,
(3) over reflexive models, the fixpoint alternation hierarchy is strict.
In Chapter 3 we discuss the modal µ-calculus on well-founded and transitive models. In extending the language by allowing fixpoints to bound also
negative occurrences of free variables and showing that it collapses on the modal
fragment, we provide a new proof of the uniqueness theorem of Bernardi, De
Jongh and Sambin and of the existence theorem of De Jongh, Sambin. For
the last one we also give a simple algorithm which shows how the fixpoint can
be computed. Chapters 2 and 3 are based on two journal papers joint with
Luca Alberucci ([3, 4]). Chapter 4 is a kind of “bridge” between the first
and the second parts of the dissertation. Indeed, in this chapter we show that
modal logic corresponds exactly to the Borel fragment of the modal µ-calculus
on transitive models. This is done by providing a bunch of equivalent effective
characterizations for the temporal logic EF on arbitrary trees. More specifically we prove that up to EF-bisimilarity, the property of being definable by
an EF formula and the property of being a Borel set coincide for languages of
both finite and infinite trees. Since we can verify that every WMSO-language is
Borel, we also immediately obtain that the EF-bisimulation invariant fragment
of weak monadic second order logic with the child relation is the logic EF. Because all these properties are proved to be equivalent also with a nice effective
algebraic characterization of EF-definability for finitely branching trees given by
Bojanczyk and Idziaszek [24], as a corollary we obtain their decidability. This
chapter is based on a joint work with Balder ten Cate [40].
The shorter Second Part of the dissertation consists of Chapters 5 and 6.
Chapter 5 introduces and discusses the two topological hierarchies that will
be investigated in the remaining chapters, as well as some useful new definitions
concerning weak automata. In Chapter 6, we introduce a new subclass of weak
10
INTRODUCTION
alternating tree automata, linear game automata, and provide an effective characterization for all the three corresponding hierarchies: index hierarchy, Borel
hierarchy and Wadge hierarchy. Moreover, we verify that for every language
recognized by these automata, the Borel rank and the Mostowski-Rabin index
coincide. This chapter is based on a conference paper joint with Jacques Duparc
and Filip Murlak [50].
In the Conclusion, we summarize the main results and indicate directions
for further research. Finally, the Appendix A contains the long combinatorial
proof of a result used in Chapter 6.
Related works
Part 1
The strictness of the fixpoint alternation hierarchy has been first proven by
Bradfield (cf. [32, 33]). Simultaneously, Lenzi [80] has proven a strictness theorem for the positive µ-calculus, that is, the fragment consisting of all formulae
such that the propositional variables appear only positively. The same question,
but restricted to full binary trees, has been independently solved by Bradfield
[34] and Arnold [7].
The question whether the modal µ-calculus hierarchy collapses on special
classes of transition systems has been addressed in various other works. A
prominent subclass, coming from Gödel-Löb logic, is the class of transitive upward well-founded frames. As shown by Visser [123] and van Benthem [13] by
using the De Jongh-Sambin fixpoint Theorem, the modal µ-calculus collapses
to its modal fragment. An analogous result via the same technique but for the
class of finite trees with the descendant relation is obtained in [39] by ten Cate,
Fontaine and Litak. This collapse also follows from the main result proved by
Bojanczyk and Walukiewicz [28].
Concerning the hierarchy on transitive frames, d’Agostino and Lenzi [44]
propose a different purely combinatorial proof which explicitly uses Theorem
2.17 of this thesis. Moreover, by using some byproducts of their proof, the
authors are able to show an unexpected behavior arising over finite transitive
frames, namely that, despite the fact that the µ-calculus and modal logic do not
coincide on finite transitive models, the bisimulation invariant fragments of first
order and monadic second order logic coincide, meaning that the µ-calculus is
included in first-order logic1 . This inclusion also follows from a characterization
of the bisimulation invariant fragment of MSO given by Dawar and Otto [46].
In this work, the authors extend the characterization given by van Benthem
[12] of modal logic as the bisimulation invariant fragment of first order logic,
and the characterization of the modal µ-calculus as the bisimulation invariant
fragment of MSO given by Janin and Walukiewicz [67], to several subclasses of
graphs, including transitive graphs, rooted graphs, finite rooted graphs, finite
transitive graphs, well-founded transitive graphs, and finite equivalence graphs.
From the above results the authors also obtain the collapse of the µ-calculus
over transitive models.
1 Note that this inclusion does not hold for the class of all transitive transition systems, a
counterexample being well-foundedness.
11
In [45] D’Agostino and Lenzi extend the study of the expressiveness of the
modal µ-calculus over some special classes of finite graphs characterized by
having strongly connected components of size bounded by a finite constant.
Notice that those classes generalize finite trees and (up to bisimulation) finite
transitive graphs. In their paper, the authors show that on the considered classes
of models, the µ-calculus collapses to the second semantical ambiguous class of
the corresponding fixpoint alternation hierarchy2. These results also generalize
an analogous result presented in [44] by the same authors but for the class of
finite transition systems where every strongly connected component has at most
one node for each possible label.
In the last years, thanks to an effort towards understanding fragments of
CTL∗ and the successful use of what are called forest algebra (cf. [29]), effective
characterizations of logics on trees have been obtained. Notably, this formalism
has been used for obtaining decidable characterizations for the classes of tree
languages definable in EF + EX [28], EF + F−1 [20, 106], BC − Σ1 (<) [26, 106],
and ∆2 (≤) [27, 106]. This approach has then been extended in the case of
the temporal logic EF on infinite finitely branching trees by Bojanczyk and
Idziaszek [24]. The decidability of the problem of knowing whether a formula of
the modal µ-calculus is equivalent to a modal formula on transitive transition
systems follows as an (almost) immediate corollary of the main result of this
last work.
We already mentioned that at the level of transition systems, thanks to the
work of Janin and Walukiewicz [67], the expressive power of the µ-calculus is
well understood. In a series of articles [63, 64, 65], Janin and Lenzi highlight
the fact that the relationship between the modal µ-calculus and monadic second
order logic is richer than expected. This is done by comparing the first levels of
the fixpoint alternation hierarchy with the bisimulation invariant fragments of
the first levels of the quantifier alternation hierarchy of MSO. More precisely,
while from [12] we already know that the bisimulation invariant fragment of
the first level of the quantifier alternation hierarchy (that is first order logic)
corresponds to the first level of the fixpoint alternation hierarchy (that is modal
logic), the authors are able to prove that this correspondence holds up to the
the level Σ2 of the monadic hierarchy of formulae3 . They also remark that this
correspondence cannot hold higher in the hierarchy. Observe that the second
level of the fixpoint alternation hierarchy of µ-formulae where the outermost
fixpoint operator is a greatest fixpoint is known to be as expressible as Büchi
tree automata [97, 66].
Part 2
Measuring the topological hardness of recognizable languages has a long tradition of research. In the case of infinite words, their understanding is almost
complete since Wagner’s 1977 paper [126]. In the same paper, Wagner was
also the first to discover remarkable relations with the index hierarchy. Sub2 Over the class of all transition systems, the second ambiguous class of the fixpoint alternation hierarchy is known to coincide with the alternation free fragment [77]. However this
does not imply that the same result must hold over the considered subclasses, leaving open the
question of whether the µ-calculus collapses to the alternation free fragment in those cases.
3 This level is given by MSO formulae of the kind ∃X · · · ∃X ∀Y · · · ∀Y φ, with φ a first
n
m
1
1
order formula.
12
INTRODUCTION
sequently, decision procedures determining an ω-regular language’s position in
both the topological and the index hierarchies were given by several authors
[75, 100, 127]. On the contrary, the situation for trees is not so satisfactory. For
example, although it is known that the index hierarchy is strict for deterministic
[127], non deterministic [96], alternating [7, 34] and also for weak alternating
tree automata [89], very little is known about the problem of computing the
minimal index of a tree language. The only case examined satisfactorily is
that of deterministic automata. On the one hand, Niwinski and Walukiewicz
[101, 102] give algorithms to compute the deterministic and nondeterministic
indices for deterministic languages, while Murlak [93] prove that for deterministic languages the Borel hierarchy and the weak index hierarchy coincide4 . On
the other hand, still Niwinski and Walukiewicz [102] show that the topological
complexity of deterministic tree languages goes much higher than that of ωregular languages. Indeed all recognizable set of infinite words are in the third
Borel ambiguous class, while deterministic automata over trees can recognize
either a language in the class Π03 in the Borel hierarchy or even Π11 -complete
languages. They are also able to show that it can be decided effectively which of
the two cases takes place. In [90] Murlak solve the missing cases, thus providing
a complete procedure calculating the position of a deterministic language in the
Borel and projective hierarchies. For what concerns the Wadge hierarchy for deterministically recognizable sets of infinite trees, Murlak [92, 93] give a complete
description of it. In particular he is able to provide an elementary procedure
to decide if one deterministic tree language is continuously reducible to another
and to show that the hierarchy has the height ω ω·3 + 3, which should be com2
pared with ω ω for regular ω-languages [127], ω ω for deterministic context-free
ω-languages [49], (ω1 CK )ω for ω-languages recognized by deterministic Turing
machines [111], or an unknown ordinal ξ > (ω1CK )ω for ω-languages recognized by
non-deterministic Turing machines, and the same ordinal ξ for non-deterministic
context-free languages [55]. For non-deterministic or alternating automata the
only results obtained are strictness theorems for various classes [32, 33, 89, 96],
and lower bounds for the heights of the hierarchies [51, 114].
It follows from the definition of the acceptance condition of non deterministic
automata and from Rabin’s Complementation Theorem that every regular set
of trees is a ∆12 -set, (cf. [107, 104]). But there are only few known results on the
complexity of non Borel regular tree languages. Simonnet [112] gives examples
of Dωn (Σ11 )-complete regular tree languages. Arnold and Niwinski [10] show
that the game tree languages form a infinite hierarchy of non Borel regular
sets of trees with regard to the continuous (Wadge) reducibility. More recently,
Finkel and Simonnet [56] investigate the topological complexity of non Borel
recognizable tree languages with regard to the difference hierarchy of analytic
sets. They prove for instance that for each integer n > 0, there is a Dωn (Σ11 )complete tree language accepted by a non-deterministic Muller tree automaton,
that a tree language recognized by an unambiguous Büchi tree automaton must
be Borel, and that almost all game languages are not in any class Dη (Σ11 ), for
η < ωω .
When studying hierarchies, the first question is to know if the considered
hierarchy is strict. However, another interesting question is to know whether
4 In the same paper Murlak propose a procedure computing for a deterministic automaton
an equivalent minimal index weak automaton.
13
the considered hierarchy also satisfies what is called the separation property,
which roughly corresponds to the fact that given two disjoint sets on a certain
level, they can be separated by a set from the lower level. In topology, it is
well known that the separation property holds for the class of analytic sets, but
fails for the class of co-analytic sets (c.f. [70]). In [11] Santocanale and Arnold
thoroughly investigate the separation property within the µ-calculus and the
index hierarchy. They show that separability fails in general starting from the
third level. For the first level, a closer look at Rabin’s original proof ([108]) of
the fact that if a set of infinite trees can be defined both by an existential and
by a universal MSO sentence then it can also be defined by a sentence of weak
monadic second order logic, reveals that the (stronger) separation property holds
for the class of Büchi recognizable languages, or equivalently for the second pi
class in the fixpoint alternation hierarchy. The missing case of level 2 is solved
by Hummel, Michalewski and Niwinski [61], by exhibiting two disjoint languages
recognized by co-Büchi tree automata which cannot be separated by any Borel
set. This result may be read as an evidence of a strong analogy between the
class of Büchi recognizable tree languages and the class of analytic set. However
in the same paper, the authors show that this analogy is not perfect. Indeed,
whereas Rabin [108] already observed that all Büchi tree languages are definable
by existential sentences of monadic second order logic, and therefore analytic,
they verify that the converse is not true. This is done by exhibiting an analytic
tree language, recognized by a parity tree automaton, but not by any Büchi
automaton.
14
INTRODUCTION
Acknowledgements
First and foremost, I want to thank my two supervisors, Jacques Duparc and
Igor Walukiewicz. Throughout the last three years, they showed unwavering
support and were always there when help was needed. Doing research under
their supervision was both a great pleasure and a huge opportunity. I benefited
greatly from their inspiring ideas, comments and suggestions, but also from the
ways in which - scientifically - they complement one another.
Others who deserve a big and warm thanks are my friends and co-authors,
with whom I spent time writing and doing research in joy, and sometimes a
mild pain. I am immensely obliged to Luca Alberucci, Balder ten Cate and
Filip Murlak. All chapters being the result of our fruitful collaborations, all
the positive outcomes of this thesis are also collectively theirs. Mistakes and
inaccuracies are only mine.
Because I was lucky enough to cross their paths during my academic life, I am
very thankful to (in chronological order): Denis Miéville, Nadine Gessler, Pierre
Joray, Arthur de Pury, Béatrice Godart, Cédric Degrange, Jean-Yves Béziau,
Alexandre Costa Leite, Jérémie Cabessa, Ramon Jansana, Ignasi Jané, Stéphane
Salaet, Miguel Angel Mota, Fabrice Correia, Christian W. Bach, Gerhard Jäger,
Thomas Strahm, Giacomo Lenzi, Giovanna D’Agostino, Johan van Benthem,
Yde Venema, Gaëlle Fontaine, Amélie Gheerbrant, Claire David, Raphael Carroy, Mikolaj Bojanczyk, Inanc Seylan and Phokion Kolaitis.
Last but not least, thanks to all my family and friends for their untiring
support and love.
Financial support from the Swiss National Science Foundation (SNSF), grant
number 100011- 116508 (Project “Topological Complexity, Games, Logic and
Automata”), is gratefully acknowledged.
Chapter 1
Logics, Games and
Automata
1.1
Trees and transition systems
In this section we introduce two kinds of structures widely used and studied in
theoretical computer science and logic: trees and transition systems.
From the practical perspective of those disciplines, trees arise in several
forms, for example as the parse trees obtained in the process of parsing a string
of symbols according to a context-free grammar or as abstract models of discrete
systems. On the other hand, it was advocated by many authors that transition
systems provide a very good framework for describing for instance the behavior
of programs. Both structures will provide semantics for the modal µ-calculus
and some of its fragments studied in this work.
1.1.1
Trees and their undressing
Given a set, or alphabet, ∆, by ∆∗ we denote the set of all finite words over
∆, by ∆+ the set of nonempty sequences and ∆ω denotes the set of all infinite
words over ∆. The symbol ε stands for the empty word. The concatenation
of a finite word v ∈ ∆∗ with a possibly infinite word w ∈ ∆∗ ∪ ∆ω is denoted
by vw. This operation is naturally generalized for any sequence of finite words.
We say that a finite word v ∈ ∆∗ is a prefix of a word w ∈ ∆∗ ∪ ∆ω if there
is a word u ∈ ∆∗ ∪ ∆ω such that vu = w. By ∆n we denote the set of words
over ∆ of length n, and by ∆<n we denote the set of all the words over ∆ of
length less than n. The set of all words over ∆ of length at most n is then the
set ∆≤n = ∆<n ∪ ∆n . Given x ∈ ∆∗ and n ∈ N, by |x| we denote the length of
the word x, and, if |x| ≥ n, by x[n] the prefix of x of length n. A set S ⊆ ∆∗
is said to be closed under prefixes if for every v ∈ S, if w is a prefix of v, then
w ∈ S.
A tree is any subset of ∆∗ closed under the prefix relation. Given a tree
T ⊆ ∆∗ , we call an element x ∈ T a node. If x is a prefix of a node y ∈ T , then
x is called an ancestor of y in T . A node that is not an ancestor of any other
node is called a leaf of the tree. If x, y ∈ T and there exists w ∈ ∆ such that
x = yw, we say that x is a child of y in T . In what follows we are interested in
15
16
CHAPTER 1.
trees whose nodes have labels from a certain set Σ. Formally, a tree over Σ, or
a Σ-tree, is any partial function t : ∆∗ → Σ such that dom(t) is a tree. By t.x
we denote the subtree of t rooted in x ∈ dom(t). A Σ-tree is called regular if,
up to isomorphism, it has only finitely many subtrees. A branch in a tree t is a
maximal chain for the prefix order in dom(t).
Those trees can have both infinite and finite branches. We call them conciliatory. A tree is called full if dom(t) = ∆∗ . Thus, assuming that the underlying
set ∆ is known and fixed, by TΣc we denote the set of all conciliatory trees over
Σ and by TΣ the set of full trees over Σ. A conciliatory tree is said to be wellfounded if it has no infinite branches. A Σ-tree t is called finitely branching if
for every node u ∈ dom(t) the set of its children is finite.
Given a full Σ-tree t, by t[n] we denote the Σ-tree given by restricting the
domain of t to ∆≤n . t[n] is called the initial Σ-tree of height n of the tree t. If
t′ corresponds to a initial Σ-tree of height n of some full tree t, we just call t′
an initial tree (of height n). Initial trees (of height n) are thus usually denoted
(n)
by t[n]. In what follows, by TΣ we denote the the set of all initial Σ-trees of
(<ω)
height n, and by TΣ
we denote the the set of all initial Σ-trees.
Given two sets L and T of conciliatory Σ-trees, the set L · T is the set of all
trees obtained by replacing in a tree t ∈ L each occurrence of a leaf by a tree
from the set T . When L is a singleton {t}, we sometimes write t · TΣ instead of
{t} · TΣ . Finally, given a node x ∈ ∆∗ , and a language L ∈ TΣ , by xL we denote
the set of Σ-trees t such that t.x ∈ L.
When ∆ = {0, 1} we call a tree binary. From now we always assume that ∆
is either N or {0, 1}.
It will be useful to relate full and conciliatory trees one to another. Different
perspectives can be adopted. Here we present a solution, given by Duparc and
Murlak [51], that will be used when playing games with full trees. Another
solution will be introduced and used in Chapter 4. The idea that will be pursue
in this chapter is that given a full tree t ∈ TΣ∪{s} , we want to skip a node
labeled by s and replace it with its leftmost child. That is, if we are in a node v
such that s = t(v), then we want to replace v with v0. But another problem is
that we may encounter an infinite sequence of nodes labeled by s. This would
keep us replacing the current node with its left child, and never get to a node
not labeled with s. In that case, the undressing of t will simply not contain
the node v. Formally, we have the following. Let t ∈ TΣ∪{s} be a full tree, the
undressing of t, denoted by U (t), is the conciliatory tree defined as follows. Let
x be the first node of t not labelled with s on the leftmost path of the tree (if
there is no such node, then U (t) is empty). Then for each v ∈ ∆∗ , consider two
possibly infinite sequences:
• w0 = ε, v0 = v,
• for vi = bv ′ , wi+1 = wi b, vi+1 = 0v ′ , if t.x(wi+1 ) = s and vi+1 = v ′
otherwise.
If vn = ε for some n, then v ∈ dom(U (t)) and U (t)(v) = t.x(wn ). Otherwise,
v ∈
/ dom(U (t)). The undressing of conciliatory trees is defined analogously as
for full trees1 .
1 Formally, given a conciliatory Σ ∪ {s}-tree t, it is enough to consider its full companion tf
over Σ ∪ {s} defined by the condition: if x ∈ dom(t), then tf (x) = t(x), otherwise tf (x) = s.
Thus, the undressing of t is defined as being the undressing of tf .
1.1. TREES AND TRANSITION SYSTEMS
17
A non-empty tree t over Σ can also be characterized as a relation structure
over the signature h<, (Pa : a ∈ Σ)i, where < is a binary relation and all the Pa
are unary relations:
Wt := hdom(t), <t , (Pat : a ∈ Σ)i
with
• <t is the child relation: <t = {(u, v) ∈ dom(t) × dom(t) : v = wi, i ∈ ∆},
and
• for all a ∈ Σ, Pat = {v ∈ dom(t) : t(v) = a}.
1.1.2
Transition systems
A transition system T over a set Σ is of the form (S, →T , λT ) where S is a non
empty set of states →T is a binary relation on S called the accessibility relation
and the function λT : S → Σ is a labeling function, associating to every state s
a label in Σ. When Σ = ℘(A), with for example A being a set of propositional
variables, it it useful to take λT as a valuation function from A into ℘(S).
A transition system T with a distinguished state s is called a pointed transition system and denoted by (T , s). T(Σ) denotes the class of all pointed
transition systems over Σ. Given any property P , by TP (Σ) we denote the
subclass of pointed transition systems over Σ satisfying the property P . In particular Tr (Σ) denotes all pointed reflexive transition systems, Tst all pointed
symmetric and transitive transition systems over Σ, Tt (Σ) all pointed transitive
transition systems and Trst (Σ) denotes all pointed transition systems over Σ
where the accessibility relation is an equivalence relation. Given any property
P , with TP f (Σ) we denote the subclass of finite pointed transition systems over
Σ satisfying the property P . For example, Ttf (Σ) denotes all finite pointed
transition systems over Σ where the accessibility relation is transitive. When
the set Σ is clear from the context, we omit to write the parameter Σ in all the
previous notations.
Let T = (S, →T , λ) be a transition system and s, s′ two states in S. A
sequence s0 , s1 , . . . , sn such that si →T si+1 , s0 = s and sn = s′ is a path of
length n connecting s to s′ . We say that s′ is reachable from s. A subset S′ ⊆ S
of the set of states is called a strongly connected component if for all s, s′ ∈ S′
we have that s′ is reachable from s. For each s by scc(s) we denote the greatest
strongly connected component which contains s if there is one and scc(s) = ∅ if
s is not contained in any strongly connected component. Note, that the notion
scc(s) is well-defined. Given a pointed transition system (T , s) and a state s′ in
it, we define the depth of s′ , dp(s′ ), to be the length of the shortest path from
s to s′ . Since parts which are non connected to the point s will be irrelevant in
the sequel we assume that all transition system are connected and, therefore,
that dp(s′ ) is defined for all s′ .
Note that we can see a tree t over Σ as a pointed transition systems (Tt , ε)
over the same set Σ by taking S = dom(t), →Tt =< and λTt = t.
Clearly, as for trees, also a transition system T over Σ can be characterized
as a relation structure over the signature h<, (Pa : a ∈ Σ)i, where < is a binary
relation and all the Pa are unary relations:
WT := hS, <T , (PaT : a ∈ Σ)i
18
CHAPTER 1.
with
• <T corresponds to the accessibility relation →S , and
• for every a ∈ Σ, PaT = {s ∈ S : λ(s) = a}.
1.2
Topology
Trees do not play only a central role in computer science but also in topology
and descriptive set theory, thus providing a natural bridge between the two
fields. In this section we introduce some basic notions and concepts that will
be used in Chapter 4 and more extensively in the whole second part of this
dissertation.
1.2.1
The Borel sets
A topological space is a set X together with a collection τ of subsets of X
containing the empty set and X, and closed under arbitrary unions and finite
intersections. The collection τ is usually called a topology on X and its elements
are called the open sets. The complements of the elements in the topology are
called closed sets. When the underlying topology is understood, we simply call
X a topological space. A base B for a topology τ on X is a family of sets in τ
such that every set in τ is an union of elements in B. A topological space X is
called second countable if it has a countable base.
A metric on a set X is a function d : X × X → R such that for every
x, y, z ∈ X:
(1) d(x, y) = 0 iff x = y
(2) d(x, y) = d(y, x), and
(3) d(x, z) ≤ d(x, y) + d(y, z)
A metric space is then a pair (X, d) where d is a metric on X. When the
underlying metric is understood, we simply call X a metric space. Let (X, d) a
metric space, x ∈ X and r > 0. We put
B(x, r) = {y ∈ X : d(x, y) < r}
and call it the open ball with center x and radius r. Clearly the collection
τ given by all subsets of X which are the union of a family of open balls of
X is a topology on X. A metric space (X, d) is said to be complete if the
limit of every sequence in X whose elements become arbitrarily close to each
other with respect to d as the sequence progresses is still in X. A topological
space whose topology is induced by a metric is called a metrizable space. It is
called completely metrizable if its topology is induced by a complete metric. A
metrizable space which is also second countable is called a Polish space.
Consider the class TΣ of all full trees2 over a finite set Σ. Let d : TΣ ×TΣ → R
the function defined by:
(
0
if t = s
d(t, s) =
−1−n
2
with n = min{|w| : t(w) 6= s(w)} otherwise.
2 The definition of the metric is the same either that T
Σ is the space obtained by taking
∆ = N or that it is the space obtained by taking ∆ = {0, 1}.
1.2. TOPOLOGY
19
Clearly d is a complete metric on TΣ .
Remark 1.1. The open balls of the metric space TΣ are given by fixing any
initial tree over Σ and then taking all possible full trees extending it. This
means that a subset L of TΣ is an open sets if there is a set L′ of initial Σ-trees
such that L = L′ · TΣ . This implies that if TΣ is the space of all full binary trees,
the space has a countable base and therefore is a Polish space.
Let (X, τ ) be a topological space. The class of Borel sets Borel(X) is the
smallest collection of subsets of X that contains the open sets and is closed
under the set-theoretical operations of countable unions and complementation.
Given two topological spaces X and Y , a function f : X → Y is said to be
continuous if for every open set O ⊆ Y , f −1 (O) ⊆ X is also open. Here, by
f −1 (O), we mean the set {x ∈ X : f (x) ∈ O}. The next well known result
states that the class of all Borel sets is closed by continuous pre-images (see
[87]):
Proposition 1.2. Let X, Y be two topological spaces and f : X → Y a continuous function. Then for every B ⊆ Y , if B ∈ Borel(Y ) then f −1 (B) ∈
Borel(X).
Note that by the previous proposition, if we have a subset B of a topological
space Y and a subset A of a topological space X which is known for not being
Borel, together with a continuous function f : X → Y such that f −1 (B) = A,
we can conclude that B is also not Borel.
The next lemma gives an example of a non Borel set of full binary trees.
Lemma 1.3 ([101]). The set Wa of all full binary trees over {a, b} where in every
branch there are only finitely many nodes with label a is not Borel.
The class of Borel sets Borel(X) of a topological space X can naturally be
spread in a hierarchy of length ω1 , called the Borel hierarchy. More precisely,
for every space X we recursively define the classes Σ0ξ (X), Π0ξ (X) and ∆0ξ (X)
as follows:
• L ∈ Σ01 (X) iff L is an open set of X,
• if 1 ≤ ξ < ω1 , L ∈ Π0ξ (X) iff L∁ ∈ Σ0ξ (X),
• if 2 ≤ ξ < ω1 , L ∈ Σ0ξ (X) iff there is a sequence hLn : n ∈ ωi of elements
S
S
of η∈ξ Π0η (X) such that L = n∈ω Ln ,
• if 1 ≤ ξ < ω1 , L ∈ ∆0ξ (X) iff L ∈ Σ0ξ (X) and L ∈ Π0ξ (X).
where for every subset L ⊆ X, L∁ = {t ∈ X : t ∈
/ L}.
0
0
By
convention,
Π
(X)
=
{X}
and
Σ
(X)
= ∅. Clearly Borel(X) =
0
0
S
S
0
0
Π
(X).
It
is
well
known
that
(see [87]):
Σ
(X)
=
ξ
ξ
ξ<ω1
ξ<ω1
Theorem 1.4. For X a topological space:
(1) Π0ξ (X) ⊆ Σ0ξ+1 (X),
(2) Σ0ξ (X) ⊆ Π0ξ+1 (X), and
20
CHAPTER 1.
(3) if Π01 (X) ⊆ Π02 (X), then
(a) Π0ξ (X) ⊆ Π0ξ+1 (X),
(b) Σ0ξ (X) ⊆ Σ0ξ+1 (X), and
(c) Π0ξ (X) ∪ Σ0ξ (X) ⊆ ∆0ξ+1 (X).
The condition Π01 (X) ⊆ Π02 (X) always holds in a metric space X, since
\
C=
{x : ∃y ∈ C, d(x, y) < (n + 1)−1 }
n∈ω
for C a closed set.
Whenever the space X is determined by the context, we omit it in the above
notations and simply write Σ0n , Π0n , and so on.
Σ01 (X)
>
>
>
>
Σ03 (X)· · ·
>
>
∆01 (X)
∆02 (X)
∆03 (X) · · ·
>
>
>
>
>
Π01 (X)
>
>
Π00 (X)
Σ02 (X)
>
Σ00 (X)
>
Π02 (X)
>
Π03 (X)· · ·
Figure 1.1: The first levels of the Borel hierarchy in the case Π01 (X) ⊆ Π02 (X).
Arrows stand for set-theoretic inclusion.
1.2.2
The Wadge game
Let X and Y be two topological spaces, and L ⊆ X and M ⊆ Y . We say that
L continuously reduces or Wadge reduces to M whenever there is a continuous
function f : X → Y such that f −1 (M ) = L. If this is the case, “topologically”
this means that L is less complicated than M and we write L ≤W M . If
L ≤W M , but M W L, then L is strictly less complicated than M , and
we write L <W M . Whenever L ≤W M and M ≤W L, we say that L and
M are Wadge-equivalent, and write L ≡W M . The sets L and M are called
incomparable when both L W M and M W L hold. In addition, a set L is
called self-dual if L ≡W L∁ , and non self-dual otherwise.
For us, the crucial observation is that, when considering full (binary or not)
trees, the relation ≤W can be characterized by a two-players infinite game with
perfect information, called the Wadge game ([124, 125]). This game is defined
as follows. Let L ⊆ TΣ and M ⊆ TΣ′ be two sets of full trees. The Wadge game
GW ((L, TΣ ), (M, TΣ′ )) is played by two players, Spoiler who is in charge of L,
and Duplicator, who is in charge of the set M . At every round, Spoiler plays
first, and both players add one more complete layer of nodes to the tree they
constructed in the previous round. Duplicator is allowed to skip her turn whenever she wishes, postponing the choice of the next layer in his tree. However,
21
1.2. TOPOLOGY
she may not do so infinitely many times in a row (otherwise the tree constructed
in the limit will not be full). Spoiler is not allowed to do so. Formally, skipping means adding an entire layer of nodes labelled by a “skip” label s ∈
/ Σ′
to the tree, and playing a full tree means, for Duplicator, playing a full tree
whose undressing is still a full tree. At the limit, Spoiler and Duplicator have
respectively produced two full trees tI ∈ TΣ and tII ∈ TΣ′ , where formally tII
is given by the undressing with respect to s of the tree produced by Duplicator
at the end of the game. Duplicator wins the game GW ((L, TΣ ), (M, TΣ′ )) iff the
condition (tI ∈ L ⇔ tII ∈ M ) holds. From this point onward, the Wadge game
GW ((L, TΣ ), (M, TΣ′ )) will be denoted by GW (L, M ) and the underlying spaces
and alphabets involved will always be assumed to be known from the context.
Along the play, the finite sequence of the previous moves of a given player is
called the current position of this player. A strategy for Spoiler is a mapping σ :
(n+1)
(<ω)
(n)
(<ω)
with the condition
such that if t ∈ TΣ′ ∪{s} then σ(t) ∈ TΣ
TΣ′ ∪{s} → TΣ
′
′′
′
that if σ(t[n]) = t [n + 1] and σ(t[n + 1]) = t [n + 2], then t [n + 1] = t′′ [n + 1],
(<ω)
(<ω)
whereas a strategy for Duplicator is a mapping σ : TΣ
\ ∅ → TΣ′ ∪{s} such
(n)
(n)
that if t ∈ TΣ′ ∪{s} then σ(t) ∈ TΣ , with the analogous monotonicity condition
stating that if σ(t[n]) = t′ [n] and σ(t[n + 1]) = t′′ [n + 1], then t′ [n] = t′′ [n].
A strategy is winning if the player following it must win, no matter what his
opponent plays.
The Wadge game was designed precisely in order to obtain that the Wadge
reduction coincides with the existence of a winning strategy for Duplicator in a
Wadge game:
Lemma 1.5 (Wadge). Let L ⊆ TΣ and M ⊆ TΣ′ be two sets of full trees. Then
L ≤W M iff Duplicator has a winning strategy in the game GW (L, M ).
Proof. Suppose Duplicator has a winning strategy σ in GW (L, M ). This strategy
(<ω)
(<ω)
naturally induces the function σ from TΣ ∪TΣ into TΣ′ ∪TΣ′ . We prove that
the restriction of this function on full trees is continuous. Let V · TΣ′ be an open
set of TΣ′ , for some set V of initial Σ′ -trees. Then σ −1 (V · TΣ′ ) = σ −1 (V ) · TΣ ,
showing that the pre-image of any open set is an open set. Moreover, the winning
condition of the Wadge game states that t ∈ M if and only if σ −1 (t) ∈ L, and
therefore σ −1 (M ) = L.
Conversely, suppose that L ≤W M . Therefore there is a continuous function
f : TΣ → TΣ′ such that f −1 (M ) = L. We describe a winning strategy for
Duplicator in the game GW (L, M ). For that purpose, consider an enumeration
{b0 , . . . , bn } of the elements of Σ′ . Since f is continuous, the sets Ai = f −1 (bi ·
TΣ′ ) form a partition of TΣ in open sets. Therefore, as long as Spoiler’s play does
not enter any of the sets Ai , Duplicator skips her turn. As soon as Spoiler’s play
enters a set Ai0 , for some i0 ≤ n, Duplicator plays on the leftmost path a node
(root of a tree, with respect to the future undressing) labelled by b0 . Notice
that since the sets Ai form a partition of TΣ , Spoiler’s play is forced to enter
one of those sets after a finite amount of time. Then, consider an enumeration
{t0 [1], . . . , tn [1], . . . } of all Σ′ -trees of height 1 whose root is labelled by bi0
and the associated sets A′i = f −1 (ti [1] · TΣ′ ), and proceed the same way. As
long as Spoiler’s play does not enter any of the sets A′i , Duplicator skips her
turn. As soon as Spoiler’s play enters an A′i1 , Duplicator plays in such a
way that the undressing of her moves is ti1 [1]. And so on and so forth. At
the end of the play, the full trees tI and tII played respectively by Spoiler
22
CHAPTER 1.
and Duplicator satisfy f (tI ) = tII . Moreover, since f −1 (M ) = L, the relation
(tI ∈ L ⇔ tII ∈ M ) holds. This strategy is therefore winning for Duplicator in
the game GW (L, M ).
The previous lemma is a very powerful result and will extensively be used
all alongs this dissertation without explicitly referring to it.
We now states some other properties of the Wadge reduction especially useful
for Part 2.
Proposition 1.6. The Wadge reduction is a pre-order.
Proof. Reflexivity is clear since the copy-cat strategy is winning for Duplicator
in GW (L, L). For the transitivity, let L, M and N three languages of full trees
such that L ≤W M, M ≤W N . Then Duplicator has two winning strategies
σ1 and σ2 in the respective games GW (L, M ) and GW (M, N ). The composition
σ2 ◦ σ1 of these two strategies is a winning strategy for Duplicator in the game
GW (L, N ), and therefore L ≤W N .
Since ≤W is a pre-order, the Wadge-equivalence relation ≡W is really an
equivalence relation. Note that clearly, for every alphabet Σ, TΣ and ∅ are
incomparable and moreover any other subset L ( TΣ Wadge-reduces them.
Some others easy properties of the Wadge-reduction are the following:
Proposition 1.7. Let L ⊆ TΣ and M ⊆ TΣ′ , then the following properties hold:
(1) L ≤W M iff L∁ ≤W M ∁ ,
(2) L and its complement L∁ are either Wadge-equivalent or incomparable,
(3) If L <W M , then M W L and M ∁ W L
Proof. The first point follows from the definition of the winning conditions of
the Wadge game and the fact that t ∈ L ⇔ t′ ∈ M iff t ∈ L∁ ⇔ t′ ∈ M ∁ . Point
two follows directly from the previous point, and point three from point two
and the fact that the Wadge-reduction relation is a pre-order.
In the preliminaries of the second part of this dissertation we will discuss
more deeply some nice and useful properties of the Wadge reduction on languages of full binary trees.
1.3
Monadic second order logics
Trees and transitions systems as relational structures can be described in monadic
second order logic with the child relation. In addition to that logic, in this section we also introduce weak monadic second order logic, which is obtained by
restricting the range of second order quantifiers to finite sets.
1.3.1
Full MSO
Let Var1 = {x, y, . . . } be the set of first order variables, and Var2 = {X, Y, . . . }
the set of second order variables. Given a finite set Σ, the set of monadic second
order (MSO) formulae over Σ is defined by the following grammar:
φ ::=< (x, y) | x = y | Pa (x) | X(x) | ¬φ | φ ∧ φ | ∃xφ | ∃Xφ
1.3. MONADIC SECOND ORDER LOGICS
23
for every a ∈ Σ, x, y ∈ Var1 and X ∈ Var2 .
As usual, we introduce implication ϕ → ψ as ¬ϕ ∨ ψ, disjunction as ¬(¬ϕ ∧
¬ψ), and the universal first order and second order quantifiers as ∀xφ ≡ ¬∃x¬φ
and ∀Xφ ≡ ¬∃X¬φ respectively. A formula φ is called a sentence if it has no
free variables.
We start with the case of trees. A valuation λ is given by a pair of functions
(
λ1 : Var1 → W ∗
λ2 : Var2 → ℘(W ∗ )
Let λ be a valuation, x a first order variable and v an element of W ∗ . The
associate valuation λ[x 7→ v] is defined for all first order variables x′ as follows:
(
v
if x′ = x,
′
λ[x 7→ v](x ) =
′
λ(x ) otherwise.
Analogously for second order variables and pairs of sequences of first and second
order variables.
We say that a valuation is consistent for a non-empty tree t if λ1 (Var1 ) ⊆
dom(t) and λ1 (Var2 ) ⊆ ℘(dom(t)). Thus, a consistent valuation for a tree t
assignes to every first-order variable an element of the domain and to each
second order variable any set of elements of the domain.
The meaning of a MSO formula φ in a structure Wt , with t non-empty, and
a consistent valuation λ for t is then defined recursively as follows:
• If φ is < (x, y), then Wt , λ φ iff <t (λ(x), λ(y))
• If φ is x = y, then Wt , λ φ iff λ(x) = λ(y)
• If φ is Pa (x), then Wt , λ φ iff λ(x) ∈ Pat
• If φ is X(x), then Wt , λ φ iff λ(x) ∈ λ(X)
• If φ is ¬ψ, then Wt , λ φ iff Wt , λ 2 ψ
• If φ is ψ ∧ γ, then Wt , λ φ iff Wt , λ ψ and Wt , λ γ
• If φ is ∃xψ, then Wt , λ φ iff there exists v ∈ W ∗ such that Wt , λ[x 7→
v] ψ
• If φ is ∃Xψ, then Wt , λ φ iff there exists a subset V ⊆ W ∗ such that
Wt , λ[X 7→ V ] ψ
We say that a formula φ is true in a structure Wt if there is a valuation λ
such that Wt , λ φ.
Given a MSO sentence φ over the alphabet Σ, the tree language defined by
φ is the set of non empty trees:
L(φ) = {t ∈ TΣc \ {∅} : ∃λ such that Wt , λ φ}
We say that a tree language L is MSO-definable if there is an MSO-formula φ
such that L = L(φ).
The case of transition systems is treated exactly in the same way as the case
of trees.
24
CHAPTER 1.
Given a MSO sentence φ over the alphabet Σ, the language of transition
systems defined by φ is the set:
L(φ) = {T ∈ T(Σ) : ∃λ such that WT , λ φ}
1.3.2
Weak MSO
Weak monadic second order logic differs from (full) monadic second order logic
in the fact that it allows to quantify not over all subsets of the domain of the
considered structure, but only on its finite subsets.
As before, let Var1 = {x, y, . . . } be the set of first order variables, and
Var2 = {X, Y, . . . } the set of second order variables. Given a finite set Σ, the
set of weak monadic second order (WMSO) formulae over Σ is defined by the
following grammar:
φ ::=< (x, y) | x = y | Pa (x) | X(x) | ¬φ | φ ∧ φ | ∃xφ | ∃f Xφ
for every a ∈ Σ, x, y ∈ Var1 and X ∈ Var2 . The concept of a sentence and of
implication, of disjunction, of universal first order quantifier and weak universal
second order quantifier are defined as expected.
As before we start by the case of trees. A weak valuation λ is given by a
pair of functions
(
λ1 : Var1 → W ∗
λ2 : Var2 → ℘fin (W ∗ )
We say that a weak valuation is consistent for a non-empty tree t if λ1 (Var1 ) ⊆
dom(t) and λ1 (Var2 ) ⊆ ℘fin (dom(t)). Thus, a consistent weak valuation for a
tree t assignes to every first-order variable an element of the domain but to each
second order variable a finite set of elements of the domain.
The meaning of a formula φ in a structure Wt , with t non-empty, and a
consistent weak valuation λ for t is then defined recursively as before, with the
new second order quantifier clause:
• If φ is ∃f Xψ, then Wt , λ φ iff there exists a finite subset V ⊆ W ∗ such
that Wt , λ[X 7→ V ] ψ
We say that a formula φ is true in a structure Wt if there is a weak valuation
λ such that Wt , λ φ.
Given a WMSO sentence φ over the alphabet Σ, the tree language defined
by φ is the set of non empty trees:
L(φ) = {t ∈ TΣc \ {∅} : ∃λ such that Wt , λ φ}
We say that a tree language L is WMSO-definable if there is an WMSO-formula
φ such that L = L(φ).
The case of transition systems is treated exactly in the same way as the case
of trees.
Given a WMSO sentence φ over the alphabet Σ, the language of transition
systems defined by φ is the set:
L(φ) = {T ∈ T(Σ) : ∃λ such that WT , λ φ}
1.4. THE MODAL µ-CALCULUS
25
Remark 1.8. The adjective “weak” is a bit misleading, since WMSO is in general not a fragment of MSO. Indeed, the class of finitely branching trees is not
definable in MSO (because, as we will see in Chapter 4, every MSO-formula that
is satisfiable on trees is true of some finitely branching tree) but is defined by the
WMSO-formula ∀x∃f X∀y(x <im y → y ∈ X), where x <im y is shorthand for
x < y ∧ ¬∃z(x < z ∧ z < y). The class of well-founded trees is definable in MSO
but not in WMSO, as will follow from results we discuss below (in particular,
the from the fact that the class of well-founded trees is not Borel). On finitely
branching trees, WMSO is strictly less expressive than MSO. This follows from
the fact that on finitely branching trees “X is a finite set” is expressed by the
MSO-formula ∀Y (∀x(Y x → Xx) ∧ ∀y(Y y → ∃z(Y z ∧ y < z)) → ¬∃y(Y y)) (“X
does not contain an infinite path”).
1.4
The modal µ-calculus
In this section we introduce the syntax and the semantics of the main object
of study of this dissertation, the modal µ-calculus. After briefly introducing
a variant of this logic, suitable for describing full binary trees, we discuss the
relation between those formalisms and monadic second order logic.
1.4.1
Syntax
The language of the modal µ-calculus, Lµ , results by adding greatest and least
fixpoint operators to propositional modal logic. More precisely, given a set Prop
of propositional variables, the collection Lµ of modal µ-formulae (or simply µformulae) is defined as follows:
ϕ ::= p | ∼p | ⊤ | ⊥ | (ϕ ∧ ϕ) | (ϕ ∨ ϕ) | 3ϕ | ϕ | µx.ϕ | νx.ϕ
where p, x ∈ Prop and x occurs only positively in ηx.ϕ (η = ν, µ), that is, ∼x is
not a subformula of ϕ. LM denotes the pure modal fragment of Lµ .
The fixpoint operators µ and ν can be viewed as quantifiers. Therefore we
use the standard terminology and notations as for quantifiers and, for instance,
free(ϕ) denotes the set of all propositional variables occurring free in ϕ and
bound(ϕ) those occurring bound. Further, we define var(ϕ) = free(ϕ)∪bound(ϕ).
If ψ is a subformula of ϕ, we write ψ ≤ ϕ. We write ψ < ϕ when ψ is a proper
subformula. sub(ϕ) is the set of all subformulae of ϕ.
Let ϕ(x) and ψ be two µ-formulae, where x occurs only positively in ϕ.
The substitution of all occurrences of x with ψ in ϕ is denoted by ϕ[x/ψ]
or sometimes simply ϕ(ψ). Simultaneous substitution of all xi by ψi (i ∈
{1, . . . , n}) is denoted by ϕ[x1 /ψ1 , . . . , xn /ψn ]. For serial substitution such as
(ϕ[x1 /ψ1 ])[x2 /ψ2 ] we often omit the parentheses and write ϕ[x1 /ψ1 ][x2 /ψ2 ].
Remark 1.9. Note, that if ϕ(x), ψ ∈ Lµ then ϕ[x/ψ] need not be a µ-formula,
for example, if we set ϕ ≡ µy.x and ψ ≡∼y then we have ϕ[x/ψ] ≡ µy. ∼y 6∈
Lµ . Nevertheless, in this paper, if nothing else mentioned, an expression like
ϕ[x/ψ] will always denote a well defined µ-formula. For a formal introduction
of substitution we refer to Alberucci [2].
The negation ¬ϕ of a µ-formula ϕ is defined inductively such that ¬p ≡∼p
and ¬(∼p) ≡ p, by using de Morgan dualities for boolean connectives and the
26
CHAPTER 1.
usual modal dualities for 3 and . For µ, ν we define
¬µx.ϕ(x) ≡ νx.¬ϕ(x)[x/¬x]
and ¬νx.ϕ(x) ≡ µx.¬ϕ(x)[x/¬x].
As usual, we introduce implication ϕ → ψ as ¬ϕ ∨ ψ and equivalence ϕ ↔ ψ as
(ϕ → ψ) ∧ (ϕ → ψ).
The fixpoint alternation depth, ad, of a formula is the number of non-trivial
nestings of alternating least and greatest fixpoints. Formally, it is defined as
follows.
Definition 1.10. Let ϕ be a µ-formula. An alternating µ-chain in ϕ of length
k is a sequence
ϕ ≥ µx0 .ψ0 > νx1 .ψ1 > · · · > µ/νxk−1 .ψk−1
where for every i < k − 1 the variable xi is free in every ψ such that ψi ≥
ψ ≥ ψi+1 . The maximum length of an alternating µ-chain in ϕ is denoted
by maxµ (ϕ). ν-chains and maxν (ϕ) are defined analogously. The alternation
depth of a µ-formula ϕ, denoted by ad(ϕ), is the maximum of maxµ (ϕ) and
maxν (ϕ). If ϕ is a purely modal formula, we set ad(ϕ) = 0.
Given a µ-formula ϕ, for all set of bound variables X ⊆ bound(ϕ), the formula ϕfree(X) is obtained from ϕ by eliminating all fixpoint operators binding a
variable x ∈ X and leaving the previously bound variables x as free occurrences.
Further, if X = {xi , . . . , xn } ⊆ bound(ϕ) then we define
ϕ−X ≡ ϕfree(X) [x1 /⊥, . . . , xn /⊥].
Example 1.11. Let ϕ be the formula µx.(3x ∨ (νy.(y ∧ µz.p ∧ 3z))), and let
X = {x, z} ⊆ bound(ϕ). Then ϕfree(X) is the formula 3x ∨ (νy.(y ∧ (p ∧ 3z))),
and ϕ−X is the formula 3⊥ ∨ (νy.(y ∧ (p ∧ 3⊥)))
We say that a variable x ∈ bound(ϕ) is well-bounded in ϕ if no two distinct
occurrences of fixpoint operators in ϕ bind x, and x occurs only once in ϕfree({x}) .
A propositional variable p is guarded in a formula ϕ ∈ Lµ if every occurrence
of p in ϕ is in the scope of a modal operator. A formula ϕ of Lµ is said to be
guarded if and only if for every subformula of ϕ of the form ηx.δ, x is guarded
in δ. The next definition is very important.
Definition 1.12. A formula ϕ of Lµ is said to be well-named if it is guarded
and every x ∈ bound(ϕ) is well-bounded in ϕ.
Notice that for all well-named3 ϕ, if x is bound in ϕ then there is exactly
one subformula ηx.δ ≤ ϕ which bounds x; this formula is denoted by ϕx .
Example 1.13. The formulae x ∧ µx.(3x ∨ p) and µx.(x ∨ y) are both non
well-named. This is because on the one hand the variable x occurs twice in
x ∧ (3x ∨ p) and on the other hand x is not guarded in µx.(x ∨ y). On the
contrary, the formulae µx.(3x ∨ νy.y) and y ∧ µx.(3x ∨ p) are well-named.
3 In the subsection of the semantics, we will see that any µ-formula ϕ is equivalent to a
well-named formula wn(ϕ).
1.4. THE MODAL µ-CALCULUS
27
If x ∈ bound(ϕ) and x is in the scope of a 3 operator in ϕx , resp. operator,
then we say that x is weakly existential in ϕ, resp. weakly universal in ϕ. If
x ∈ bound(ϕ) and x is in the scope only of 3 operators in ϕx , resp. operators,
then we say that x is existential in ϕ, resp. universal in ϕ. Let ϕ(x) be a µformula. If x is free and occurs only positively in ϕ, then we define ϕn (x) for
all n inductively such that ϕ1 (x) = ϕ(x) and such that
ϕk+1 (x) ≡ ϕ[x/ϕk (x)].
We define ϕn (⊤) = ϕn [x/⊤], and analogously for ϕn (⊥).
In order to make proofs by induction on the structure of a formula, we now
introduce a measure for the syntactical complexity of formulae of the modal
µ-calculus. This is done by the so called rank function, which assigns to each
formula a well-chosen ordinal number.
Definition 1.14. The rank, rank(ϕ), of a formula ϕ is an ordinal number defined inductively as follows:
• rank(p) = rank(∼p) = 1
• rank(△ α) = rank(α) + 1 where △∈ {, 3}
• rank(α ◦ β) = max{rank(α), rank(β)} + 1 where ◦ ∈ {∧, ∨}
• rank(ηx.α) = sup{rank(αn (x)) + 1 ; n ∈ N} where η ∈ {ν, µ}.
The fact that the definition of rank, originally introduced in [3], terminates
is shown in [5] (see also [2]). It is an easy exercise to show that for all formulae
ϕ we have that rank(ϕ) = rank(¬ϕ).
Example 1.15.
(1) The rank of the formula 3(p ∨ 3q) is 5. In general, the rank of any
modal formula is a finite ordinal.
(2) Let ϕ1 = µx.(3x ∨ p). The rank of ϕ1 is ω, while the rank of 3ϕ1 is ω + 1.
(3) Let ϕ2 = µx.µy.((3x ∨ p) ∧ y). The rank of ϕ2 is ω 2 .
(4) Let ϕ3 = νz.µw.(z ∧ (w ∧ ϕ2 )). The rank of ϕ3 is ω 2 · 2.
(5) Let ϕ4 = p ∧ (⊤ ∨ νx.(3x ∨ ϕ3 ). The rank of ϕ4 is ω 2 · 2 + ω + 2.
Remark 1.16. In [5], Alberucci and Krähenbühl are able to show that for every
µ-formula ϕ, rank(ϕ) < ω ω and that for every ordinal ξ < ω ω there is a formula
whose rank is exactly ξ. Moreover they provide an algorithm to compute the
rank of any formula by primitive recursion.
The axioms and inference rules below define the deduction system Koz, introduced by Kozen in [73]. As usual we write Koz ⊢ ϕ if there is a derivation of
ϕ in the system presented below.
Axioms: All classical propositional tautologies, the Distribution Axiom from
modal logic
((α → β) ∧ α) → β
28
CHAPTER 1.
and the Fixpoint Axioms
ηx.ϕ(x) ↔ ϕ(ηx.ϕ(x)),
η ∈ {µ, ν}.
Inference Rules: Beside the classical Modus Ponens
α α→β
β
and the Necessitation Rule
we have the Induction Rule
α
,
α
ϕ → α[x/ϕ]
.
ϕ → νx.α
It can be shown that the fixpoint axioms can be replaced by the following weaker
axioms:
νx.ϕ(x) → ϕ(νx.ϕ(x)) and ϕ(µx.ϕ(x)) → µx.ϕ(x).
1.4.2
Semantics
The semantics of modal µ-calculus is given by trees or transition systems over
℘(Prop). Since the semantics given by trees can be seen, mutatis mutandis, as
a special case of the semantics given by transition systems, we only present this
last one.
Let (S, →T , λT ) be a transition system with valuation function λT : S →
℘(S), p a propositional variable and S′ a subset of states S; the associated
valuation λT [p 7→ S′ ] is defined for all propositional variables p′ as follows:
(
S′
if p′ = p,
T
′
′
λ [p 7→ S ](p ) =
λT (p′ ) otherwise.
Given a transition system T = (S, →T , λT ), then T [p 7→ S′ ] denotes the transition system (S, →T , λT [p 7→ S′ ]). This notions are generalized straightforwardly
to λ[x1 7→ S1 , . . . , xn 7→ Sn ] and T [x1 7→ S1 , . . . , xn 7→ Sn ]. Given a transition
system T , the denotation of ϕ in T , kϕkT , that is, the set of states satisfying
a formula ϕ is defined inductively on the structure of ϕ. Simultaneously for all
transition systems we set
• kpkT = λ(p) and k ∼pkT = S \ λ(p) for all p ∈ Prop,
• kα ∧ βkT = kαkT ∩ kβkT ,
• kα ∨ βkT = kαkT ∪ kβkT ,
• kαkT = {s ∈ S | ∀t((s →T t) ⇒ t ∈ kαkT )},
• k3αkT = {s ∈ S | ∃t((s →T t) ∧ t ∈ kαkT )},
S
• kνx.αkT = {S′ ⊆ S | S′ ⊆ kα(x)kT [x7→S′ ] }, and
T
• kµx.αkT = {S′ ⊆ S | kα(x)kT [x7→S′ ] ⊆ S′ }.
1.4. THE MODAL µ-CALCULUS
29
We say that a pointed transition system (T , s) is a model of a µ-formula if and
only if s ∈ kϕkT . By kϕk we denote the class of all models of ϕ and by kϕkP
the class of all models of ϕ with property P . For example, by kϕkt , we denote
the class of all transitive models of ϕ. Whenever kϕk = T, we write |= ϕ. For
a formula ϕ(x) and set of states S′ ⊆ S we sometimes write kϕ(S′ )kT instead of
kϕ(x)kT [x7→S′ ] . When clear from the context we use kϕ(x)kT for the function
(
℘(S) → ℘(S)
kϕ(x)kT :
S′ 7→ kϕ(S′ )kT .
By the Tarski-Knaster Theorem (c.f. [119]), kνx.α(x)kT is the greatest fixpoint
and kµx.α(x)kT the least fixpoint of the operator kα(x)kT .
We say that a class L ⊆ TP of transition systems is definable by a µ-formula,
or simply µ-definable, if there exists ϕ ∈ Lµ such that L = kϕkP .
Thanks to Walukiewicz, we have a completeness theorem for the modal µcalculus.
Theorem 1.17 ([128]). For all µ-formulae ϕ we have that
|= ϕ
if and only if
Koz ⊢ ϕ.
The next two lemmas state some basic properties of denotations. Their
proofs are standard and left to the reader.
Lemma 1.18. For all transition systems T = (S, →T , λT ) and all formulae ϕ
we have that
(1) k¬ϕkT = S \ kϕkT ,
(2) kηx.ηy.ϕ(x, y)kT = kηx.ϕ(x, x)kT , where η ∈ {µ, ν},
(3) kνx.ϕ(x)kT = kϕ(⊤)kT , if all occurrences of x are not guarded,
(4) kµx.ϕ(x)kT = kϕ(⊥)kT , if all occurrences of x are not guarded.
Lemma 1.19. Let ϕ, α, αi , β, βi , ψ, ψi ∈ Lµ be well-named µ-formulae, where
i ∈ {1, . . . , k}. For all transition systems T the following holds:
(1) If free(ψi ) ∩ bound(ϕ) = ∅ for all i ∈ {1, . . . , k} then
kϕ[x1 /ψ1 , . . . , xk /ψk ]kT = kϕkT [x1 7→kψ1 kT ,...,xk 7→kψk kT ] .
(2) If ψ ≤ ϕ and xi ∈ free(ψ) ∩ bound(ϕ), with i = 1, . . . , k, then
kψ[x1 /ϕx1 , . . . , xk /ϕxk ]kT = kψkT [x1 7→kϕx1 kT ,...,xk 7→kϕxk kT ] .
(3) If free(ψi ) ∩ bound(α) = free(ψi ) ∩ bound(β) = ∅ and xi ∈ free(α) ∩ free(β)
for all i ∈ {1, . . . , k} and for every transition system T we have that
kαkT = kβkT
then, for every transition system T we have that
kα[x1 /ψ1 , . . . , xk /ψk ]kT = kβ[x1 /ψ1 , . . . , xk /ψk ]kT .
30
CHAPTER 1.
(4) Let free(αi ) ∩ bound(ϕ) = free(βi ) ∩ bound(ϕ) = ∅ and let xi ∈ free(ϕ)
occur positively in ϕ, where i = 1, . . . , k. If for every transition system T
we have that
kαi kT ⊆ kβi kT , for every i ∈ {1, . . . , k},
then we have that
kϕ[x1 /α1 , . . . , xk /αk ]kT ⊆ kϕ[x1 /β1 , . . . , xk /βk ]kT .
To conclude, we prove a useful lemma showing that a formula and its wellnamed companion have the same meaning and the same alternation depth.
Lemma 1.20. For all µ-formulae ϕ there is a well-named formula wn(ϕ) such
that for all T we have kϕkT = kwn(ϕ)kT and ad(ϕ) = ad(wn(ϕ)).
Proof. We have just to verify that the construction of wn(ϕ) given by parts 2
to 4 of Lemma 1.18 does not increase the alternation depth of the formula. But
this is straightforward.
Given Lemma 1.21 and Lemma 1.20, we can assume that wn is a function
associating to every formula ϕ a well-named formula wn(ϕ) which has the same
alternation depth and the same denotation in every transition system. Moreover,
the next lemma shows that wellnaming iterated formulae which are already wellnamed does not affect the rank. It follows by the fact that since ϕ is well-named
the well-named formula corresponding to ϕn (⊤) is given by simply renaming
bound variables.
Lemma 1.21. For all well-named formulae ϕ such that x ∈ free(ϕ) appears only
positively and all n ∈ N we have that
rank(ϕn (⊤)) = rank(wn(ϕn (⊤))).
Similarly for ⊥.
From now on, if nothing else is mentioned, we assume that all µ-formulae
are well-named.
1.4.3
Another formalism for the binary tree
In this subsection we introduce a slightly modified language for the modal µcalculus interpreted over full binary trees. The only difference is that the underlying modal language allows two different existential modalities: one for the left
child and one for the right child. The reason of introducing this version of the
modal µ-calculus is because it has an exact counterpart in terms of automata
that will be studied in details in the second part of this dissertation.
Formally, given a set Prop of propositional variables, the collection of formulae of the bi-modal µ-calculus is defined as follows:
ϕ ::= p | ¬ϕ | (ϕ ∨ ϕ) | (ϕ ∧ ϕ) | h0iϕ | h1iϕ | µx.ϕ | νx.ϕ
where p, x ∈ Prop and x occurs only positively in ηx.ϕ (η = ν, µ), that is, x is
the scope of an even number of negations.
1.4. THE MODAL µ-CALCULUS
31
Mutatis mutandis, the alternation depth of a formula, all the other standard
syntactical notions and the semantics (with the obvious interpretations for the
two existential modalities) on full binary trees are defined as for the “normal”
modal µ-calculus, as well as the definitions of evaluation game, alternation free
fragment, and of the syntactical and semantical fixpoint hierarchies that will
follow in the next sections. Note that for every formula a semantically equivalent formula in positive normal form (the only negated subformulae are free
variables) can be obtained by repetitive uses of de Morgan laws, the usual equivalences connecting fixpoints and the obvious equivalences: ¬hiiϕ ≡ hii¬ϕ, for
i = 0, 1. Finally, it is worth noticing that on full binary trees the standard
modal µ-calculus can be seen as a fragment of the bi-modal µ-calculus just by
defining 3ϕ ≡ h0iϕ ∨ h1iϕ.
1.4.4
MSO vs µ-calculus
Since the work of Niwiniski [96, 97], it was known that the modal µ-calculus
with two existential modalities corresponds exactly to monadic second order
logic4 on binary trees.
But if we look to a transition system as representing the behavior of a process, we usually do not want to distinguish between operationally equivalent
such structures. Since bisimulation relation is often considered to be the finest
equivalence relation which is interesting in this context, this means that a specification language should not distinguish between bisimilar transition systems.
Properties of trees or transition systems are naturally expressed in quantified
languages. But such languages can distinguish between bisimilar structures.
This is for example the case of first-order logic, but interestingly not of modal
logic. In fact, van Benthem [12] proved that modal logic express exactly the first
order properties on transition systems which are invariant under bisimulation,
meaning that from an operational point of view, all first order properties which
make sense are modal. Unfortunately, first order logic is often not expressive
enough. Indeed, it cannot express properties like liveness and safety, contrary
to a second order language. So, what about monadic second order logic? And
where does the modal µ-calculus stand? The question has been solved by Janin
and Walukiewicz. In [66] they show that all monadic second order properties of
transition systems which are bisimulation invariant are exactly the ones which
are expressible in the modal µ-calculus, meaning that this logic is the “right”
weakening of second order logic.
So, let’s first recall when two transition systems are said to be bisimilar.
A bisimulation relation between two pointed transition systems (T0 , s0 ) and
(T1 , s1 ) over an arbitrary set Σ is a relation R ⊆ ST0 × ST1 such that (s0 , s1 ) ∈ R
and for every (s′0 , s′1 ) ∈ R:
• λT0 (s′0 ) = λT1 (s′1 ),
• whenever s′0 →T0 s′′0 , for some s′′0 ∈ ST0 , then there exists s′′1 ∈ ST1 such
that s′1 →T1 s′′1 and (s′′0 , s′′1 ) ∈ R, and
• whenever s′1 →T1 s′′1 , for some s′′1 ∈ ST1 , then there exists s′′0 ∈ ST0 such
that s′0 →T0 s′′0 and (s′′0 , s′′1 ) ∈ R.
4 With
the child relation and the order on siblings.
32
CHAPTER 1.
Two pointed transition systems over Σ are said to be bisimilar if there exists a
bisimulation relation between them.
The fine-structure of the bisimulation relation suggests the following notion
of a bisimulation game. There are two players, Bob and Anne. Bob claims that
the two pointed transition systems (T0 , s0 ) and (T1 , s1 ) over Σ are different, while
Anne says they are bisimilar. The game proceeds in rounds. At the beginning
of each round, the state in the game is a pair of states (s′0 , s′1 ) ∈ ST0 × ST1 . A
round is played as follows. If the label of the two states determining the current
position of the play are different, then the game stops and Bob wins. Otherwise
first Bob selects one of the transition systems Ti and a →Ti -successor of s′i , for
i = 0, 1. Then Anne selects a →T1−i -successor of s′1−i in the other structure T1−i
and the round is finished. A new round is played with the position updated to
(s′′0 , s′′1 ). If a player cannot choose a successor when it is her turn in a round, she
loses. If Anne can survive for infinitely many rounds, she wins. It is then easy
to verify that (T1 , s1 ) and (T2 , s2 ) are bisimilar iff Anne has a winning strategy
in the corresponding bisimulation game.
Given a class C of pointed transition systems over Σ, C is said to be closed
under bisimulation, or bisimulation closed, if whenever (T , s) ∈ C and (T ′ , s′ )
is bisimilar to (T , s), then (T ′ , s′ ) ∈ C. Janin and Walukiewicz characterization
of the µ-calculus is therefore the following:
Theorem 1.22 ([66]). A bisimulation closed class of transition systems is definable in MSO iff it is definable by a µ-formula.
Since for all µ-formulae φ, kφk is closed under bisimulation, it turns out that
the previous theorem means that properties definable in MSO which are closed
under bisimulation are exactly the one definable with a formula of the modal
µ-calculus.
1.5
The fixpoint alternation hierarchy
What gives the modal µ-calculus its great expressive power is not simply the
use of fixpoint operators, but the possibility of nesting (alternate) greatest and
least fixpoints in a “non trivial” way. This phenomenon is important from a
practical point of view, as the alternation depth of a µ-formula is what appears
to generate the computational complexity of model-checking (cf. sections 1.6
and 1.7). From a theoretical point of view, the classification naturally raises
questions about the expressive power of the classes. In particular, the question
whether the expressiveness for the modal µ-calculus is somewhat “bounded”,
that is just a bounded number of alternation are needed to express all possible
definable properties.
In this section we study alternation in more detail by describing what are
called the syntactical and semantical hierarchies for the modal µ-calculus.
Let Φ ⊆ Lµ . For η ∈ {ν, µ}, η(Φ) is the smallest class of formulae such that:
• Φ, ¬Φ ⊂ η(Φ);
• If ψ(x) ∈ η(Φ) and x occurs only positively, then ηx.ψ ∈ η(Φ);
• If ψ, ϕ ∈ η(Φ), then ψ ∧ ϕ, ψ ∨ ϕ, 3ψ, ψ ∈ η(Φ);
33
1.5. THE FIXPOINT ALTERNATION HIERARCHY
• If ψ, ϕ ∈ η(Φ) and free(ψ) ∩ bound(ϕ) = ∅ then ϕ[x/ψ] ∈ η(Φ)
With the help of this definition, we introduce the syntactical hierarchy for the
modal µ-calculus. For all n ∈ N, we define the class of µ-formulae Σµn and Πµn
inductively as follows:
• Σµ0 := Πµ0 := LM ;
• Σµn+1 = µ(Πµn );
• Πµn+1 = ν(Σµn );
• ∆µn = Σµn ∩ Πµn
S
S
It is clear that Lµ = n∈ω Σµn = n∈ω Πµn . Moreover from the definitions above,
we can easily prove that this hierarchy is strict, that is to say, for every n ∈ ω
and Σµn ∪ Πµn ( Πµn+1
Σµn ∪ Πµn ( Σµn+1
All Σµn and Πµn classes form the syntactical modal µ-calculus hierarchy, also
called the syntactical fixpoint alternation hierarchy.
Σµ1
>
>
∆µ2
>
>
∆µ1
>
Πµ1
>
>
Σµ0
>
>
>
>
||
Σµ2
Σµ3 · · ·
∆µ3 · · ·
>
>
>
Πµ0
>
Πµ2
>
Πµ3 · · ·
Figure 1.2: The syntactical fixpoint hierarchy. Arrows stand for set-theoretic
inclusion.
It can be asked what is the relation between levels of the syntactical modal
µ-calculus hierarchy and the fixpoint alternation depth. This relation is summarized in the following result:
Proposition 1.23 ([99]). Let k ∈ N and ϕ ∈ Lµ . Then:
(1) ad(ϕ) ≤ k iff ϕ = φ[x1 /ψ1 , . . . , xl /ψl ] for some formula φ ∈ Σµ0 and
formulae ψi ∈ Σµk ∪ Πµk ,
(2) the formula ϕ belongs to Σµk iff ad(ϕ) ≤ k and there is no subformula
νx.ψ ≤ ϕ such that maxν (νx.ψ) = k
(3) the formula ϕ belongs to Πµk iff ad(ϕ) ≤ k and there is no subformula
µx.ψ ≤ ϕ such that maxµ (µx.ψ) = k
34
CHAPTER 1.
We therefore obtain that the alternation depth of ϕ ∈ Lµ is equivalently
given by
ad(ϕ) = inf{k : ϕ ∈ ∆µk+1 }.
In particular this means that the fixpoint alternation free fragment, that is the
class of µ-formulae whose alternation depth is 1, corresponds to the second
ambiguous syntactical class ∆µ2 .
The semantical modal µ-calculus hierarchy over T, also called the semantical
µT
fixpoint alternation hierarchy over T, consists of all ΣµT
n and Πn , which are
classes of pointed transition systems defined inductively as follows:
µ
ΣµT
n = {kϕk : ϕ ∈ Σn }
µ
ΠµT
n = {kϕk : ϕ ∈ Πn }
As usual, the ambiguous classes are defined by
µT
µT
∆µT
n := Σn ∩ Πn .
The semantical modal µ-calculus hierarchy over TP , for any property P (like
being a transitive model), is defined analogously.
ΣµT
1
>
>
>
>
>
>
∆µT
···
3
>
>
>
ΣµT
···
3
>
>
>
ΣµT
0
>
∆µT
2
∆µT
1
||
ΣµT
2
>
ΠµT
0
ΠµT
1
>
ΠµT
2
>
ΠµT
···
3
Figure 1.3: The semantical fixpoint hierarchy. Arrows stand for set-theoretic
inclusion.
The strictness of the semantical fixpoint hierarchy over all transition systems
has been first proven by Bradfield (cf. [32, 33]). Simultaneously, Lenzi in
[80] has proven a strictness theorem for the positive µ-calculus, that is, the
fragment consisting of all formulae such that the propositional variables appear
only positively.
Theorem 1.24 (Bradfield). The semantical modal µ-calculus hierarchy over
T is strict.
The same question, but restricted to full binary trees, has been independently solved by Bradfield [34] and Arnold5 [7].
5 In his original formulation, Arnold’s result concerns the analogous hierarchy for alternating
automata, the so-called index hierarchy, a hierarchy that will be introduced and discussed in
section 1.9. The strictness of the fixpoint hierarchy on binary trees can then be derived
from this result thanks to the equivalence between alternating automata and the bi-modal
µ-calculus, cf. section 1.8.
35
1.6. PARITY GAMES
A classical result by Rabin [108] states that if a set of infinite trees and
its complement are both definable by an existential and by a universal MSO
sentence, then these sets are definable in weak monadic second order logic. In
the language of the modal µ-calculus, the theorem asserts the equality, on infinite trees, between the second ambiguous class of the corresponding semantical
fixpoint alternation hierarchy and the class of tree models definable by an alternation free formula [8]6 . This result is surprisingly no more true as we climb
higher in the hierarchy, as shown by Arnold and Santocanale [11].
From now on, when we write about the modal µ-calculus hierarchy, resp. the
fixpoint alternation hierarchy, we always mean the semantical modal µ-calculus
hierarchy, resp. the semantical fixpoint alternation hierarchy.
Example 1.25. It is instructive to have a look at two typical µ-formulae. The
first formula express the property of “always eventually p”
νx.(µy.(p ∨ 3y)) ∧ x).
Indeed, it says that from any node of a model, we can reach a node where
p holds. Since this formula is in Πµ1 , this kind of property can be expressed
without any alternation. Moreover, it can be shown that this formula cannot
be reduced to a purely modal formula. The second formula defines the property
of “there is a path where p holds infinitely often”
νx.µy.((p ∨ 3y)) ∧ 3x).
It can be verified that the alternation is really needed, that is, that the class of
µT
models of this formula is in ΠµT
2 \ Σ2 .
1.6
Parity games
Since its appearance in the verification community, it was clear that new techniques were required for the understanding of the complexity of the modal µcalculus. That’s when parity games make their remarkable entry into the arena.
By replacing old methods in proof and techniques but also by becoming a central topic of investigation in their own right, they are now an essential tool in
the study of this formalism. In the present and next sections parity games and
their link with the µ-calculus are presented and discussed.
Formally, a game G is defined in terms of an arena A and a winning condition
W . In our case an arena is simply a bi-partite graph A = hV0 , V1 , Ei, where
V0 ∩ V1 = ∅ and the edge relation, or set of moves, is E ⊆ (V0 ∪ V1 ) × (V0 ∪ V1 ).
Let V = V1 ∪ V2 be the set of vertices, or positions, of the arena. Given two
vertices a, b ∈ V , we say that b is a successor of a, if (a, b) ∈ E. The set of all
successors of a is sometimes denoted by aE or E(a). We say that b is reachable
from a if there are a1 , . . . , an ∈ V such that a1 = a, an = b and for every
0 < i < n, ai+1 ∈ ai E.
A play in the arena A can be finite or infinite. In the former case, the play
is a non empty finite path π = a1 . . . an ∈ V + such that for every 0 < i < n,
ai+1 ∈ ai E and an E = ∅. In the last case, the play consists in an infinite path
π = a1 . . . an · · · ∈ V ω with ai+1 ∈ ai E for every i > 0. Thus a finite or infinite
6 This
identity can then be lifted from trees to transition systems [77].
36
CHAPTER 1.
play in a game can be seen as the trace of a token moved on the arena by two
Players, Player 0 and Player 1, in such a way that if the token is in position
a ∈ Vi , then Player i has to choose a successor of a where to move the token.
The set of winning conditions W is a subset of V ω . Thus, given a game
G = (A, W ) a play π is winning for Player 0 iff
(1) if π is finite, then the last position an of the play is in V1 ,
(2) if π is infinite, then it must be a member of W .
A play is winning for Player 1 if it is not winning for Player 0. In this
framework we are interested in what is called a parity winning condition. This
winning condition is nowadays recognized as the most fundamental one of all
winning conditions for two-player infinite games. In particular, as we will see
in the next section, it captures exactly what is needed to evaluate formulae of
the modal µ-calculus. Formally, given a set of vertices V , we assume a coloring
or ranking function Ω : V → ω such that Ω[V ] is bounded. Then, the set W of
winning conditions is defined as the set of all infinite sequences π such that the
greatest priority appearing infinitely often in Ω(π) is even 7 .
Let A be an arena. A strategy for Player i is simply a function σi : V ∗ Vi →
V , with i = 1, 2. A prefix a1 . . . an of a play is said to be compatible or consistent
with σi iff for every j with 1 ≤ j < n and aj ∈ Vi , it holds that σi (a1 . . . aj ) =
aj+1 . A finite or infinite play is compatible or consistent with σi if each of its
prefix which is in V ∗ Vi is compatible with σi . The strategy σi is said to be a
winning strategy for Player i on W if every play consistent with σi is winning
for Player i. A position a ∈ V is winning for Player i in the parity game G iff
there is a strategy σ for Player i such that every play compatible with σ which
starts from a is winning for Player i. A winning strategy σ is called memoryless
if σ(a1 . . . an ) = σ(b1 . . . bn ), when an = bn .
Ther are two basic questions about games that usually need to be answered.
(1) (Determinacy) Does every game has a winner?
(2) How difficult is it to determine who wins a (finite) game (if there is any
winner) ?
The first question has a positive answer. That is, for parity games we have
a memoryless determinacy result.
Theorem 1.26 ([53, 89]). In a parity game, one of the Players has a memoryless
winning strategy from each vertex.
Having in mind this theorem, in the sequel we assume that all winning
strategies are memoryless, that is, a winning strategy in a parity games for
Player 0 is a function σ : V0 → V , analogously for Player 1.
The decision problem of determining the winner of a parity game is formally
given by:
(WINS) given a finite parity game G, determine whether or not Player 0 has
a winning strategy in the game G
7 A parity game is called a weak parity game if, as a winning condition, we say that Player
0 wins (either a finite or an infinite play) if and only if the greatest priority occurring in the
play is even.
37
1.7. EVALUATION GAMES
This problem is clearly P-hard, and moreover we have that8 :
Theorem 1.27 ([116, 69]).
• WINS is solvable in time
O dm(
n
⌊ d2 ⌋
d
)⌊ 2 ⌋
where n is the number of vertices, m is the number of edges, and d is the
maximum priority in a parity game
• WINS is in UP ∩ co-UP,
A big open problem is to know whether WINS is in P or not.
1.7
Evaluation games
Using the standard semantics on transition systems is not the only way, and
probably not the most transparent way, of interpreting the modal µ-calculus. It
is for instance possible to associate a meaning to formulae in terms of games. In
this section we will see, given ϕ ∈ Lµ and a pointed transition system (T , s0 ),
how to determine the corresponding parity game E(ϕ, (T , s0 )), called also the
evaluation game of ϕ over (T , s0 ). Thanks to the a “game-theoretical” version
of what is usually called the Fundamental Theorem of the semantic of the modal
µ-calculus, first proved by Emerson and Street, the two semantics are shown to
be equivalent.
Recall that T = (S, →T , λT ). The arena of E(ϕ, (T , s0 )) is the triple hV0 , V1 , Ei
which is defined recursively such that
hϕ, s0 i ∈ V
(recall that V = V0 ∪ V1 ) and such that if hψ, si ∈ V then we distinguish the
following cases:
• If ψ ≡ (¬)p and p ∈ free(ϕ). In this case we set Ehψ, si = ∅ and
(
s ∈ λT (ψ) if ψ ≡ p
hψ, si ∈ V1 iff
s 6∈ λT (ψ) if ψ ≡ ¬p.
• If ψ ≡ x and x ∈ bound(ϕ). In this case we set
(hψ, si, hϕx , si) ∈ E
and we have
hψ, si ∈ V0 iff x is a µ-variable.
• If ψ ≡ α ∧ β then we have hψ, si ∈ V1 , and if ψ ≡ α ∨ β then we have
hψ, si ∈ V0 . In both cases it holds that
(hψ, si, hα, si) ∈ E and (hψ, si, hβ, si) ∈ E
8 The
class UP is the complexity class of decision problems solvable in polynomial time
on a non-deterministic Turing machine with at most one accepting path for each input. UP
contains P and is contained in NP.
38
CHAPTER 1.
• If ψ ≡ α then we have hψ, si ∈ V1 , and if ψ ≡ 3α then we have
hψ, si ∈ V0 . In both cases it holds that
(hψ, si, hα, s′ i) ∈ E for all s′ such that s →T s′ .
• If ψ ≡ νx.α then we have hψ, si ∈ V1 , and if ψ ≡ µx.α then we have
hψ, si ∈ V0 . In both cases it holds that
(hψ, si, hα, si) ∈ E.
We complete the definition of the parity game E(ϕ, (T , s0 )) by defining the
(partial) priority function Ω : V → ω. The function is first defined on states of
the form hηx.δ, si ∈ V , where η ∈ {µ, ν}. In this case we have that:

ad(ηx.δ) if η = µ and ad(ηx.δ) is odd, or



η = ν and ad(ηx.δ) is even;
Ω(hψ, si) =
ad(ηx.δ)
−
1
if
η
= µ and ad(ηx.δ) is even, or



η = ν and ad(ηx.δ) is odd.
For a state of the form hx, si, where x ∈ bound(ϕ), we set
Ω(hx, si) := Ω(hϕx , si).
where ϕx is the unique subformula ηx.δ ≤ ϕ which bounds x.
For all the other states hα, si we distinguish two cases. If there is a least
formula ηx.δ ∈ sub(ϕ) such that ηx.δ > α we set
Ω(hα, si) := Ω(hηx.δ, si).
If there is no such formula then we set
(
1
Ω(hα, si) =
min{Ω(ηx.δ) : ηx.δ ≤ ϕ}
if ϕ is a modal formula,
else.
It can easily be seen that if there is a formula ηx.δ > α then there is also a
least one. Therefore, the second case refers to subformulae α of ϕ which can
not be regenerated by a fixpoint application in a parity game. In the following
we simply write min Ω and max Ω instead of min{Ω(hα, si) : hα, si ∈ V } and of
max{Ω(hα, si) : hα, si ∈ V }.
Recall that if the play π is finite, Player 0 wins iff the last vertex of the play
belongs to V1 , and if the play π is infinite, Player 0 wins iff the greatest priority
appearing infinitely often even.
Theorem 1.28 ([116]). (T , s) ∈ kϕk iff Player 0 has a winning strategy for
E(ϕ, (T , s)).
This result can be seen as the “game-theoretical version” of what is usually
called the Fundamental Theorem of the semantic of the modal µ-calculus.
Example 1.29. Evaluation game E(νx.((p ∨ ⊥) ∧ x), (T , s1 )). The transition
system T is as in Figure 1.4, that is, it has states {s1 , s2 , s3 } and p holds in s1
and s2 , and the accessibility relation is as depicted in Figure 1.4.
39
1.7. EVALUATION GAMES
{p}
s1
|
||
||
|
|~ |
{p}
/ s3
s2
J
Figure 1.4:
h((p ∨ ⊥) ∧ x), s1 i
OOO
OOO
OOO
OO'
V
q
qqq
q
q
q
x qq
q
h(p ∨ ⊥) ∧ x, s3 i
h(p ∨ ⊥) ∧ x, s2 i
hp ∨ ⊥, s3 i
hx, s3 i
h(p ∨ ⊥), s2 i
`
UUUU
UUUU
UUUU
UUUU
U*
hx, s2 i
ν
}
hp, s3 i
h⊥, s3 i
ν
h((p ∨ ⊥) ∧ x), s3 i
hp, s2 i
$
h⊥, s2 i
h((p ∨ ⊥) ∧ x), s2 i
Figure 1.5:
In Figure 1.5, we have the arena of E(νx.((p ∨ ⊥) ∧ x), (T , s1 )). In order
to simplify the picture we identified vertices of the form hνx.((p ∨ ⊥) ∧ x), si
with the vertices of the form h((p ∨ ⊥) ∧ x), si. Note, that this does not
change essentially the evaluation game. Further, the graph given by the nondotted edges represents the part of the arena which can be reached by a play
given the strategy of Player 0 where he chooses, if there is the possibility, the
non-dotted instead of the dotted move. Note, that it is a winning strategy. It
is then easy to verify that νx.((p ∨ ⊥) ∧ x) is valid if for all reachable states
in a transition system we have that either, the state is terminal, or, p holds in
the state.
The proof of the following lemma follows from the proof in Emerson and
Street [116] of Theorem 1.28.
Lemma 1.30. Let T = (S, →T , λT ) be a transition system and ϕ(x1 , . . . , xk )
be a formula where all xi occur positively. Let σ be a strategy for Player 0 in
E(ϕ(x1 , . . . , xk ), (T , s)). Suppose that for all vertices of the form hxi , s′ i which
are reachable by σ we have that s′ ∈ Ai ⊆ S, with i = 1, . . . , k. Then σ
can be converted into a winning strategy for Player 0 in the evaluation game
E(ϕ(x1 , . . . , xk ), (T [x1 7→ A1 , . . . , xk 7→ Ak ], s)).
A first immediate consequence of Theorem 1.28 is a complexity bound for
the model-checking problem of the modal µ-calculus. Formally, the problem is
the following:
(Model-Checking) given a finite pointed transition system (T , s) and a formula ϕ ∈ Lµ , determine whether or not (T , s) ∈ kϕk
40
CHAPTER 1.
From Theorem 1.27, this problem is in UP ∩ co-UP and is solvable in time
O dm(
n
⌊ d2 ⌋
d
)⌊ 2 ⌋
where n and m are numbers depending on (T , s) and ϕ, and d is the alternation
depth of ϕ.
Given a parity game E(ϕ, (T , s)) for a formula ϕ we define the pointed game
transition system T (E(ϕ, (T , s))) = ((S, →T , λT ), s0 ) such that the states S
are the vertices V and the distinguished state s0 = hϕ, si, and such that the
transition relation →T is the edge relation E of the parity game. If ad(ϕ) = n
then the valuation λT is specified for the new propositional variables
{ci : 0 ≤ i ≤ n} ∪ {di : 0 ≤ i ≤ n}.
For all ψ ∈ sub(ϕ) we define our valuation for these propositional variables such
that
λT (di ) = {hψ, si : hψ, si ∈ V0 and Ω(hψ, si) = i} and
λT (ci ) = {hψ, si : hψ, si ∈ V1 and Ω(hψ, si) = i}.
In the following we introduce the game formulae and show that with them it is
possible to test the existence of a winning strategy for Player 0 in an evaluation
game.
Definition 1.31. For all n ≥ 1 we define the Σµn game formula WΣµn such that:
W
Σµ
n
:≡
(
Wn−1
Wn−1
µxn−1 .νxn−2 . . . . νx0
i)
i=0 (di ∧ 3xi ) ∨
i=0 (ci ∧ x
Wn
Wn
µxn .νxn−1 . . . . µx1
i=1 (di ∧ 3xi ) ∨
i=1 (ci ∧ xi )
n even
n odd
The Πµn game formula WΠµn is defined such that:
W
Πµ
n
:≡
(
Wn
Wn
νxn .µxn−1 . . . . µx1
i=1 (di ∧ 3xi ) ∨
i=1 (ci ∧ xi )
Wn−1
Wn−1
νxn−1 .µxn−2 . . . . νx0
i=0 (ci ∧ xi )
i=0 (di ∧ 3xi ) ∨
n even
n odd
For n = 0 we define
WΣµ0 :≡ WΠµ0 :≡ WΣµ1 .
It is clear from definition that for all n ≥ 1 we have that WΣµn ∈ Σµn and
WΠµn ∈ Πµn .
Proposition 1.32 ([53, 130]). Let G an arbitrary parity game. Assume that
min Ω ∈ {0, 1} and max Ω = n. We have that if n is even (resp. odd):
(a) if min Ω = 0 then Player 0 has a winning strategy for G if and only if
T (G) ∈ kWΠµn+1 k (resp. T (G) ∈ kWΣµn+1 k),
(b) if min Ω = 1 then Player 0 has a winning strategy for G if and only if
T (G) ∈ kWΠµn k (resp. T (G) ∈ kWΣµn k)
From Proposition 1.32 and the definition of an evaluation game, it follows
immediately that:
1.8. AUTOMATA FOR THE MODAL µ-CALCULUS
41
Corollary 1.33. Let ϕ be a Πµn -formula (resp. a Σµn -formula) and let (T , s)
be an arbitrary pointed transition system. We have that Player 0 has a winning
strategy for E(ϕ, (T , s)) if and only if T (E(ϕ, (T , s))) ∈ kWΠµn k (resp. if and
only if (T , s) ∈ kWΣµn k).
Therefore, by applying Proposition 1.28 and Corollary 1.33, we have the
following result:
Corollary 1.34. Let ϕ be a Πµn -formula (resp. Σµn -formula) and let (T , s) be
an arbitrary pointed transition system. We have that
(T , s) ∈ kϕk if and only if T (E(ϕ, (T , s))) ∈ kWΠn k (resp. (T , s) ∈ kWΣn k).
1.8
Automata for the modal µ-calculus
Another way of looking at µ-formulae is to consider automata. The kind of
automata we introduce differ slightly depending on whether we are considering
full binary trees or more generally arbitrary transition systems. After presenting
both cases, we state the expected correspondences with the corresponding logical
formalism (bi-modal µ-calculus for binary trees and standard modal µ-calculus
for transition systems).
We refrain from giving a thorough introduction to automata on words and
trees. For a comprehensive introduction to automata theory, we refer to the
books [104, 57] and to the survey [118]. Throughout this section we assume
that Prop is finite.
1.8.1
Automata on binary trees
Recall that full binary trees are total functions t : {0, 1}∗ → Σ with a prefix
closed domain. From now on in this subsection TΣ denote the space of full
binary trees over Σ.
We choose to work with automata having the parity condition as acceptance
condition. An alternating parity tree automaton over a finite input alphabet
Σ A = hQ, Q∃ , Q∀ , qI , δ, Ωi consists of a finite set Q of states partitioned into
existential states Q∃ and universal states Q∀ , an initial state qI , a transition
relation δ ⊆ Q × Σ × {ε, 0, 1} × Q and a priority function Ω : Q → ω. We
σ,d
can assume that ι ∈ {0, 1}.Sometimes we write q −−→ q ′ when q ′ ∈ δ(q, σ, d), or
σ
q−
→ q ′ , q ′′ when (q ′ , q ′′ ) ∈ δ(q, σ).
The run of the alternating automaton A on an input infinite binary tree t ∈
TΣ is defined in terms of a parity game. More precisely, consider an alternating
automaton A and an infinite binary tree t ∈ TΣ . The corresponding parity game
G(A, t) is then defined as follows.
• the set V0 is {0, 1}∗ × Q∃
• the set V1 is {0, 1}∗ × Q∀
• from each vertex (v, q) and for each (q ′ , a) ∈ δ(q, t(v)), ((v, q), (va, q ′ )) ∈
E,
• for every vertex (v, q), Ω((v, q)) = Ω(q).
42
CHAPTER 1.
We say that A accepts t iff player 0 has a winning strategy in the parity game
G(A, t). The language recognized by A, denoted L(A) is the set of trees accepted
by A. We call a tree language L regular if L = L(A), for some alternating tree
automaton A. For every state q which is not the initial state qI of the automaton
A, by Aq we denote the automaton corresponding exactly to A except the fact
that the initial state now is q and not qI .
A weak alternating parity tree automaton A is an alternating parity automaton, satisfying the condition that if a state q is reachable from the state q ′ in
the graph of the automaton, then Ω(q ′ ) ≤ Ω(q)9 .
Because of the next result, we are assured of an exact correspondence of the
expressive power of alternating automata and the modal µ-calculus with two
existential modalities on binary trees:
Theorem 1.35 ([96, 53]). For every set of full binary trees L, the following two
statements are equivalent:
• there is an alternating automaton A such that L = L(A),
• there is a bi-modal µ-formula ϕ such that {(Tt , ε) : t ∈ L} = kϕk
This leads to the consequence that over binary trees, MSO, the bi-modal
µ-calculus and alternating automata have the same expressive power.
Regular languages are closed under union, intersection and complementation. Thus, for every pair of alternating automata A and B, by A ⊓ B (resp.
A ⊔ B), we denote the automaton recognizing L(A) ∩ L(B) (resp. L(A) ∪ L(B)),
and by A we denote the the automaton recognizing the language L(A)∁ . Note
that the previous closure property holds also when considering weakly recognizable languages only.
1.8.2
Automata on transition systems
The modal µ-calculus was originally defined over arbitrary transition systems
and not confined to binary trees. It is therefore interesting to know which is
the automata-theoretic counterpart of µ-formulae on those structures. In [66],
Janin and Walukiewicz define a notion of an automaton, called µ-automaton,
that operates on transition systems and that corresponds exactly to the modal
µ-calculus.
Note that an analogous equivalent characterization of µ-formulae in terms
of alternating parity automata admitting runs over arbitrary transition systems
is given by Wilke in [132].
A µ-automaton A over finite input alphabet Σ is a tuple hQ, q0 , δ, Ωi where Q
is a finite set of states, q0 ∈ Q is the initial state, δ : Q × Σ → ℘℘(Q) is the transition function, and Ω : Q → N is a bounded parity function. Given a pointed
transition system (T , s0 ) with λ : S → Σ, an A-game in (T , s) with starting position (s0 , q0 ) is played between two players, Duplicator and Spoiler. The game
9 Another way of characterizing weak alternating automata is in terms of weak parity game.
More precisely, we can say that a weak alternating parity tree automaton is defined exactly
as an alternating parity automaton, except that the run is given by a weak parity game.
Call a weak alternating automaton defined in this alternative way a second-weak alternating
automaton. Then it can then be shown (cf. [91]) that a tree language is recognized by a
weak alternating automaton of index [ι, κ] iff it is recognized by a second-weak alternating
automaton of index [ι, κ] (cf. section 1.9 for the definition of the index of a parity automaton).
1.9. THE MOSTOWSKI-RABIN INDEX HIERARCHY
43
is defined recursively as follows: if we are in position (s, q) ∈ S × Q, Duplicator
has to make a move. She chooses a marking m : Q → ℘({s′ ∈ S : s →T s′ }) and
then a description D in δ(q, λ(s)). If s′ ∈ m(q), we say that s′ is marked with q.
The marking and the description have to satisfy the two following properties.
Firstly, if q ′ ∈ D, there exists a successor s′ of s that is marked with q ′ . Secondly, if s′ is a successor of s there exists q ′ ∈ D such that s′ is marked with q ′ .
After Duplicator’s choice of a marking m, Spoiler plays a position (s′ , q ′ ) such
that s′ ∈ m(q ′ ).
If a play is finite, we say that the player who cannot make a move loses. An
infinite play (s0 , q0 ), (s1 , q1 ), . . . is won by Duplicator if the greatest element of
{Ω(q) : q appears infinitely often in q0 , q1 , . . . } is even. We say that a pointed
transition system (T , s0 ) is accepted by A if Duplicator has a winning strategy
in the A-game in (T , s0 ) with starting position (s0 , q0 ).
The set of pointed transition system over Σ accepted by A is denoted by
L(A). We say that a set of transition systems L is recognized by A if L = L(A).
The correspondence between logic and automata is then assured by the following theorem:
Theorem 1.36 ([66]). For every µ-automaton A over ℘(Prop), there is a µformula ϕ such that L(A) = kϕk. Conversely, for every µ-formula ϕ, there is a
µ-automaton A over ℘(Prop) such that L(A) = kϕk.
1.9
The Mostowski-Rabin index hierarchy
As for the modal µ-calculus, there is a natural hierarchy resulting from the
structure of automata with the parity acceptance condition, the hierarchy of
the Mostowski-Rabin indices of parity automata. This hierarchy orders languages according to the nesting of positive and negative conditions checked by
the recognizing automaton. It has two main versions: weak, when considering
only weak alternating automata; and strong, when considering all alternating
automata. This hierarchy is believed to reflect the inherent computational complexity of the language, and therefore has attracted a lot of attention encouraged
by the expectations of the verification community [32, 33, 75, 96, 100, 101, 102].
Formally, the (Mostowski-Rabin) index of the automaton A is the pair [ι, κ],
where ι is the minimal value and κ is the maximal value of the priority function
of A. An automaton with index [ι, κ] is often called a [ι, κ]-automaton. For an
index [ι, κ] we shall denote by [ι, κ] the dual index, i.e. [0, κ] = [1, κ + 1] and
[1, κ] = [0, κ − 1]. Let us define the following partial order on indices:
[ι, κ] ⊑ [ι′ , κ′ ]
iff
either {ι, . . . , κ} ⊆ {ι′ , . . . , κ′ } or {ι + 2, . . . , κ + 2} ⊆ {ι′ , . . . , κ′ }
In other words, one index is smaller than another if and only if it uses less
priorities. This means that dual indices are not comparable. The (MostowskiRabin) index hierarchy for a certain class of automata consists of ascending sets
(levels) of languages recognized by [ι, κ]-automata.
44
CHAPTER 1.
[0, 1]
>
>
[0, 2]
>
[0, 1] ∩ [1, 2]
>
[0, 3] · · ·
>
>
[0, 2] ∩ [1, 3]
[0, 3] ∩ [1, 4] · · ·
>
>
>
[1, 2]
>
>
[1, 3]
>
[1, 4] · · ·
Figure 1.6: The Mostowski-Rabin index hierarchy. Arrows stand for settheoretic inclusion. We omit the first two trivial levels [0, 0] and [1, 1].
In the case of alternating tree automata (or µ-automata, when dealing with
transition systems) the hierarchy is called the strong index hierarchy. For weak
alternating automata, the hierarchy is called the weak index hierarchy.
As usual, the first fundamental question about the hierarchy concerns its
strictness, i.e. the existence of languages recognized by a [ι, κ]-automaton, but
not by a [ι, κ]-automaton. Because of a nice exact correspondence between
the index of an µ-automata and the alternation depth of the equivalent µformula (cf. [132]), Bradfield’s result on the strictness of the fixpoint alternation
hierarchy on arbitrary transition systems implies the strictness of the strong
index hierarchy on those structures. For full binary trees, we have already seen
that the strong index hierarchy was shown to be strict by Arnold in [7] and
independently by Bradfield in [34]. It turns out that Arnold’s proof can also be
applied to weak alternating automata, showing therefore the strictness of the
weak index hierarchy, a result already obtained by Mostowski [88] but using
a reduction to a hierarchy of weak monadic second-order quantifiers formerly
examined by Thomas [117]. A direct proof of the strictness of the weak index
hierarchy can also be found in [101].
In the remainder of the section we give another proof of Arnold’s results for
binary trees in terms of Wadge games by following the path taken by Arnold
and Niwinski in a recent work [10].
Consider the alphabet Σ[ι,κ] = {0, 1} × {ι, . . . , κ} with ι ∈ {0, 1} and ι ≤ κ.
Then, to every full binary tree t over Σ[ι,κ] , we associate a parity game G(t) as
follows: a node v in the tree is a position for player 0 iff the first component of
the node is 0, and the rank of the node corresponds to its second component.
The set W[ι,κ] corresponds to the class of trees in TΣ[ι,κ] for which Player 0
has a winning strategy in the corresponding parity game G(t). For every index
[ι, κ], the set W[ι,κ] is called the the game language of index [ι, κ]. All those
languages are regular. It is enough to consider to automaton A[ι,κ] defined by:
• the set of states is the set {qσk : σ ∈ Σ[ι,κ] , k ∈ {0, 1}} ∪ {qI } partitioned
by
k
– Q∃ = {q(i,j)
∈ Q : i = 0, k ∈ {0, 1}} ∪ {qI } and
k
– Q∀ = {qi,j
∈ Q : i = 1, k ∈ {0, 1}},
• the transition relation δ is defined by
1.9. THE MOSTOWSKI-RABIN INDEX HIERARCHY
45
– δ(qI , σ, ε) = qσ0 , for all σ ∈ Σ[ι,κ] ,
– δ(qσ0 , σ, d) = qσ1 for all σ ∈ Σ[ι,κ] and d ∈ {0, 1},
– δ(qσ1 , σ ′ , ε) = qσ0 ′ , for all σ, σ ′ ∈ Σ[ι,κ] .
k
– the priority function is defined by Ω(qI ) = ι and Ω(q(i,j)
) = j, for
every (i, j) ∈ Σ[ι,κ] and k ∈ {0, 1}, meaning that the index of A[ι,κ]
is [ι, κ].
Clearly L(A[ι,κ] ) = W[ι,κ] .
Moreover, it is easy to see that every parity game G(A, t) can be effectively
encoded from the root into a full binary tree tW ∈ W[ι,κ] , with [ι, κ] being the
index of the automata A, in such a way that Player 0 has a winning strategy
in G(A, t) iff she has a winning strategy in G(tW ). Moreover, this “encoding” is
continuous. More precisely, we have the following:
Proposition 1.37. Let A be any alternating parity tree automaton of index
[ι, κ] over Σ. Then L(A) ≤W W[ι,κ] .
′
Proof. Let W[ι,κ]
be the class of all full finitely branching trees over Σ[ι,κ] for
which Player 0 has a winning strategy in the corresponding parity game G(t)
′
Then, the winning strategy for Duplicator in the Wadge Game GW (L(A), W[ι,κ]
)
is given by just playing at every turn the arena of the parity game G(A, t), where
t is the finite tree precisely played by Spoiler at that moment. Therefore we
′
have that L(A) ≤W W[ι,κ] . But clearly W[ι,κ]
≤W W[ι,κ] . Indeed, the winning
′
strategy for Duplicator in the Wadge Game GW (W[ι,κ]
, W[ι,κ] ) is the following.
First, given the (finite) finitely branching tree t : {ǫ, 0, 1, . . . , n} → Σ, with
dom(t) = {ǫ, 0, 1, . . . , n}, let f (t) : {0, 1}∗ → Σ be the binary encoding of t
given by f (t)(ε) = f (t)(1j ) = t(ε), with 1 ≤ j < n, f (t)(1n ) = t(n) and
f (t)(1j 0) = t(j), for 0 ≤ j < n. We say that ǫ ∈ dom(f (t)) corresponds to
ǫ ∈ dom(t), 1n ∈ dom(f (t)) corresponds to n ∈ dom(t) and that for every 1 ≤
j < n, 1j 0 ∈ dom(f (t)) corresponds to j ∈ dom(t). We then define inductively
for every finite finitely branching tree t and for every node v ∈ dom(t), the binary
encoding f (t) of t and which is the (unique) node in dom(f (t)) corresponding
to v.
′
Consider the following strategy for Duplicator in GW (W[ι,κ]
, W[ι,κ] ):
(1) at the first round, copy Spoiler’s move,
(2) for every round n ≥ 2, for every terminal node v of the tree constructed
by Spoiler after round n − 1, if t is the tree constructed after Spoiler’s turn
at round n, then replace the terminal node corresponding to v with the
binary encoding of t.v.
By definition of the binary encoding, this is a well-defined winning strategy for
Duplicator. By transitivity of ≤W we therefore obtain that L(A) ≤W W[ι,κ] .
It is worth noticing that the game languages witness the strictness of fixpoint
alternation hierarchy, and therefore of the strong index hierarchy (cf. [33, 7]).
This result was recently strengthened by Arnold and Niwinski in [10]. In their
paper, the authors show that the class of game languages form a hierarchy with
respect to the Wadge reducibility. In the remaining part of the section we give
46
CHAPTER 1.
a very short proof of the the same results by exploiting the game-theoretical
characterization of Wadge reducibility. But before this, we explicitly establish
the link between the Wadge hierarchy and the index hierarchy.
Lemma 1.38. If the Wadge hierarchy of tree game languages is strict, then the
index hierarchy of alternating parity tree automata is strict too.
Proof. Suppose that the Wadge hierarchy of for tree game languages is strict,
but that the index hierarchy for alternating parity tree automata collapses.
Assume it collapses to the [ι, n] level. Consider the automata recognizing the
game tree language W[ι,n+1] . By hypothesis, there is an alternating tree automaton A of index [ι, n] recognizing the same language. By Proposition 1.37,
we are able to construct a winning strategy for Spoiler in the Wadge Game
GW (W[ι,n+1] , W[ι,n] ). But this contradicts the strictness of the Wadge hierarchy
of tree game languages.
We can now give another proof of Arnold and Niwinski’s result [10]. First,
we have the following lemma:
Lemma 1.39. Spoiler has a winning strategy in both GW (W[1,n+1] , W[0,n] ) and
GW (W[0,n] , W[1,n+1] ), for every n ≥ 0.
Proof. We only prove that Spoiler has a winning strategy in the Wadge game
GW (W[1,n+1] , W[0,n] ), the other case being identical. This is done by describing
the winning strategy for Spoiler in this game. As a first move, Spoiler plays a
finite binary tree over {[1, 1]}. Then the strategy goes as follows:
(1) if Duplicator skips, then for every terminal node add two children labelled
by [1, 1]
(2) otherwise, at every terminal node, add two dual copies of what Duplicator
has already played as successors.
Clearly this is a winning strategy for Spoiler in GW (W[1,n+1] , W[0,n] ).
Proposition 1.40 ([10]). The Wadge hierarchy for tree game languages is strict.
Proof. It is trivial to verify that, for every n, Duplicator has a winning strategy in both GW (W[0,n] , W[0,n+1] ) and GW (W[0,n] , W[1,n+2] ), and dually in both
GW (W[1,n+1] , W[1,n+2] ) and GW (W[1,n+1] , W[0,n+1] ). Therefore, if we show that
Spoiler has a winning strategy in GW (W[0,n+1] , W[0,n] ) and GW (W[1,n+2] , W[0,n] )
(and in the dual case) we are done. We only prove that Spoiler has a winning
strategy in GW (W[0,n+1] , W[0,n] ), the other cases being identical. We do this by
describing the winning strategy for Spoiler in this game. As a first move, she
plays an finite binary tree over the alphabet {(1, 0)}. Then the strategy goes as
follows:
(1) if Duplicator skips, then for every terminal node add two nodes labelled
by (1, 0).
(2) otherwise, by Lemma 1.39, apply the winning strategy for Spoiler in the
Wadge game GW (W[1,n+1] , W[0,n] ).
Clearly this is a winning strategy for Spoiler in GW (W[0,n+1] , W[0,n] ).
1.10. SUMMARIZING REMARKS
47
By applying Proposition 1.40 to Lemma 1.38, we can therefore obtain an alternative, very easy, proof of the strictness of the index hierarchy for alternating
tree automata.
Proposition 1.41. The index hierarchy for alternating parity tree automata is
strict.
Mutatis mutandis, by considering the weak counterpart of game languages,
the same argument yields the strictness of the index hierarchy of weak alternating parity automata.
1.10
Summarizing remarks
The modal µ-calculus is an extension of modal logic with greatest and least fixpoint operators. This formalism, introduced in Section 1.4, is used to described
properties of transition systems (labelled directed graphs) or trees. However
using the standard semantics is not the only way, and probably not the most
transparent way, of interpreting the µ-calculus. It is for instance possible to associate a meaning to formulae in terms of parity games (Section 1.6), highlighting
the fact that the automata counterpart of this logic is given by alternating parity automata (Subsection 1.8.1 in the case of binary trees, and Subsection 1.8.2
in the case of transition systems).
We have also seen that another way of speaking about properties of trees and
transition systems consists in using monadic second order logic10 (Section 1.3).
But curiously, at least at first sight, we do not necessarily gain so much with this
logic in term of expressiveness when compared to the modal µ-calculus. Indeed,
on the one hand, since the work of Niwiniski [96, 97] we know that on binary
trees the modal µ-calculus with two existential modalities corresponds exactly
to monadic second order logic11 . On the other hand Janin and Walukiewicz [66]
have shown that all bisimulation invariant monadic second order properties of
transition systems are exactly the ones which are expressible in the µ-calculus.
We are interested here in understanding the expressive power of some fragments the modal µ-calculus, and thus, since they are equivalent to the latter,
of subclasses of alternating automata. A first, natural, measure of complexity
is given by what is called the fixpoint alternation hierarchy (Section 1.5), resp.
the Mostowski-Rabin (index) hierarchy for parity automata (Section 1.9). This
hierarchy reflects the depth of nesting of greatest and least fixpoints, resp. of
positive and negative conditions. In the case of transition systems, the fixpoint
hierarchy has been proved to be strict by Bradfield [32, 33]. Since Bradfield [34]
and Arnold [7], this hierarchy is also known to be strict on binary trees. Chapter
2 and Chapter 3 study the behavior of the fxipoint alternation hierarchy of the
µ-calculus when we restrict the class of the considered models.
Infinite trees also play a central role in topology and descriptive set theory.
In these settings, sets of trees can be compared via continuous, or Wadge, reductions (Subsection 1.2.2) or be classified by using for instance the classical Borel
hierarchy (Subsection 1.2.1). It is this therefore very natural to ask whether
the two points of view (combinatorial vs topological) have some connections,
10 Notice that in this dissertation we have decided to use monadic second order logic with
only the child relation, meaning that, as models of this logic, trees are seen as unordered trees.
11 With the child relation and the order on siblings.
48
CHAPTER 1.
or if they measure two different kinds of complexity. While this issue will be
more extensively treated in the Second Part of the thesis, and partly already
in Chapter 4, a first known link is presented in Section 1.9. Bradfield [33] and
Arnold [7] have indeed observed that the so called game tree languages testify
of the strictness of fixpoint, resp. index, hierarchy. More precisely, they proved
that those languages form a strict hierarchy for the alternation of fixpoints,
resp. the Mostowski-Rabin indices. This result has then been strengthened by
Arnold and Niwinski [10], who have shown that the class of game languages
form a hierarchy with respect to the Wadge reducibility. In order to exemplify
the use of Wadge Games in proofs, in the last part of this introductory chapter
we presented an easy proof by way of Wadge Games of the previous theorem
of Arnold and Niwinski and of the fact that it almost immediately implies the
strictness of the index hierarchy. In Chapter 6, the two points of view – topological and in terms of the combinatorial complexity of the recognizing automaton
– are studied and discussed for a restricted class of alternating parity automata
capturing a very weak form of alternation.
Part I
The µ-Calculus on
Restricted Classes of
Models
49
Chapter 2
The Fixpoint Hierarchy on
Reflexive, Transitive, and
Transitive-Symetric Models
This chapter is based on a joint work with Luca Alberucci [3].
2.1
Preliminary remarks
Many natural properties such as “there is an infinite path” can be expressed
by a modal µ-formula. Further, most such properties are given by formulae
with alternation depth two. Nevertheless, it is mathematically interesting to
see whether the expressive power of the modal µ-calculus increases with the
alternation depth. If this is the case then we have a strict hierarchy otherwise
we have a collapse at some point.
We already know that the semantical fixpoint alternation hierarchy over
arbitrary transition systems is strict. Having seen the strictness over arbitrary
transition systems, it can be asked whether this hierarchy remains strict for
restricted classes of transition systems such as those that are reflexive or those
that are transitive. In the case of transitive systems, to our knowledge, the
first attempt to answer this question was presented by Lenzi in [81]. There,
he shows that on transitive frames every Büchi automaton is equivalent to a
co-Büchi automaton, and conversely1. This implies that over transitive frames
the modal µ-calculus collapses to the level of Büchi automata (and to co-Büchi
automata). Since, for instance, well-foundedness is not definable in the modal
fragment, the hierarchy is non trivial. Thus, since over arbitrary graphs the
intersection of Büchi and co-Büchi automata corresponds to the alternationfree fragment, Lenzi conjectured that the full modal µ-calculus collapses to the
alternation-free fragment ([82]). It is interesting to note that Visser has shown in
[122] that in the case of reflexive and transitive models, where well-foundedness
is false and therefore can be expressed by a modal formula, the non-triviality
1 A complete proof of this fact, extended to the class of finite simple graphs (a class which
contains - modulo bisimulation - the class of finite transitive graphs) can be found in [44].
51
52
CHAPTER 2.
of the fixpoint hierarchy is testified by the formula stating the existence of an
infinite path alternately labelled with p, ¬p, p, ¬p, etc.
In this chapter we answer positively Lenzi’s conjecture for the class of all
transitive systems by giving an explicit syntactical translation of the full modal
µ-calculus into the alternation-free fragment. This result is first showed for finite transition systems and then generalized, by proving a finite model theorem,
to all transitive systems. We also verify, again by giving an explicit syntactical
translation, that if we add symmetry to transitivity the modal µ-calculus collapses to the purely modal fragment. Further, by adapting Arnold’s proof for
the general case, we show that the hierarchy remains strict over reflexive frames.
In the next section some useful finite model theorems are proved. In Sections
2.3 and 2.4 the collapse of the fixpoint hierarchy over transitive-symmetric and
over transitive systems are proved. In Section 2.5 we finally prove the strictness
of the hierarchy over the class of all reflexive transition systems.
2.2
Finite model theorems
In this section finite model theorems for the modal µ-calculus over the class of
all reflexive and over the class of all transitive transition systems are proved.
Let us first state the well-known finite model theorem for general transition
systems.
Theorem 2.1 ([74, 116]). For all modal µ-formulae ϕ for which there is a transition system T and a state s in T such that s ∈ kϕkT , there is a finite transition
system T F and a state sF such that sF ∈ kϕkT F .
2.2.1
Finite model theorem for reflexive transition systems
Let ϕ be a µ-formula. By induction on the structure of ϕ we define the formula
ϕref as follows:
• (∼)pref ≡ (∼)p,
• (α ◦ β)ref = αref ◦ βref where ◦ ∈ {∧, ∨},
• (α)ref = αref ∧ αref ,
• (3α)ref = 3αref ∨ αref , and
• (ηx.α)ref = ηx.αref where η ∈ {µ, ν}.
The next Lemma is by induction on the structure of the formula.
Lemma 2.2. Let T be a transition system and let T ref be its reflexive closure.
For all µ-formulae ϕ the following holds
s ∈ kϕref kT
if and only if
s ∈ kϕkT ref .
With the help of this lemma we can easily prove the finite model property
for reflexive transition systems.
53
2.2. FINITE MODEL THEOREMS
Theorem 2.3. For all modal µ-formulae ϕ for which there is a reflexive transition system T and a state s in T such that s ∈ kϕkT there is a finite reflexive
transition system T F and a state sF such that sF ∈ kϕkT F .
Proof. Let ϕ be a µ-formula and T a reflexive transition system with a state s
such that s ∈ kϕkT . Since T is reflexive we have that T = T ref and therefore
by Lemma 2.2 we have that
s ∈ kϕref kT .
By the general Finite Model Theorem 2.1 we get that there is a finite transition
system T F and a state sF such that
sF ∈ kϕref kT F .
If we define T F ref to be the reflexive closure of T F by applying again Lemma
2.2 we get
sF ∈ kϕkT F ref
and we have found the finite reflexive model and a state in it satisfying ϕ.
2.2.2
Finite model theorem for transitive transition systems
Let ϕ be a µ-formula. By induction on the structure of ϕ we define the formula
ϕtr as follows:
• (∼)ptr ≡ (∼)p,
• (α ◦ β)tr = αtr ◦ βtr where ◦ ∈ {∧, ∨},
• (α)tr = νx.(αtr ∧ x),
• (3α)tr = µx.3(αtr ∨ x), and
• (ηx.α)tr = ηx.αtr where η ∈ {µ, ν}.
As in the reflexive case, the next Lemma is proved by induction on the
structure of the formula.
Lemma 2.4. Let T be a transition system and let T tr be its transitive closure.
For all µ-formulae ϕ the following holds
s ∈ kϕtr kT
if and only if
s ∈ kϕkT tr .
By using Lemma 2.4, mutatis mutandis, the proof of the finite model property for transitive transition systems is exactly the same as for Theorem 2.3.
Theorem 2.5. For all modal µ-formulae ϕ for which there is a transitive transition system T and a state s in T such that s ∈ kϕkT there is a finite transitive
transition system T F and a state sF such that sF ∈ kϕkT F .
54
2.3
CHAPTER 2.
The transitive and symmetric case
In this section, we prove the collapse of the semantical modal µ-calculus hierarchy over Tst to the purely modal fragment. Let us begin with the following
easy result. Because the successors of two nodes belonging to the same strongly
connected component of a transitive transition system are the same, we immediately obtain that:
Lemma 2.6. Let T be a transitive transition system and let s′ ∈ scc(s). For all
µ-formulae ϕ we have that
s ∈ k △ ϕkT
if and only if
s′ ∈ k △ ϕkT
where △∈ {, 3}.
The next theorem assures us that over transitive and symmetric transition
systems, fixpoints can be eliminated.
Theorem 2.7. Let T be a transitive and symmetric transition system. For every
well-named µ-formula ϕ, we have that
kνx.ϕ(x)kT = kϕ(ϕ(⊤))kT .
Proof. The ⊆ inclusion is clear. For the ⊇ inclusion, define A = kϕ(ϕ(⊤))kT ;
by definition of greatest fixpoint it is enough to show that we have
A ⊆ kϕ(A)kT .
(2.1)
First recall that we assume that νx.ϕ(x) is well-named. This means that in the
formula ϕ(x) the variable x is in the scope of a modal operator and occurs only
once in ϕ. Therefore, we can assume that ϕ is of the form β(△ α(x)) where
△∈ {3, }. Moreover we have that △ α(x) and △ α(ϕ(x)) occur only once in
the formula tree of ϕ(ϕ(x)). Let s ∈ A, by Proposition 1.33 there is a winning
strategy σ for Player 0 in the evaluation game E(ϕ(ϕ(x)), (T [x 7→ S], s)). Let
π be an arbitrary play consistent with σ. If π reaches a vertex of the form
h△ α(x), s′ i then the same play reaches a vertex of the form h△ α(ϕ(x)), s′′ i.
Since σ is a winning strategy for Player 0 by Proposition 1.33 we have that
s′′ ∈ k △ α(ϕ(x))kT [x7→S]
and s′ ∈ k △ α(x)kT [x7→S] .
Since T is transitive and symmetric it clearly holds that s′′ ∈ scc(s′ ) and, by
applying Lemma 2.6, we have
s′ ∈ k △ α(ϕ(x))kT [x7→S] .
Hence, we have shown that for all plays π consistent with σ, if π reaches a vertex
of the form h△ α(x), s′ i then, by Theorem 1.28, there is a winning strategy for
Player 0 in the evaluation game E(△ α(ϕ(x)), (T [x 7→ S], s′ )). A fortiori, this
implies that if π reaches a vertex of the form h△ α(x), s′ i then there is a winning
strategy σh△α(x),s′ i for Player 0 in E(△ α(x), (T [x 7→ kϕ(x)kT [x7→S] ], s′ )). Therefore, since kϕ(x)kT [x7→S] ⊆ S, the strategy σ ∗ given by following σ but switching
to the corresponding σh△α(x),s′ i when a position of the form h△ α(x), s′ i is
reached, is winning for Player 0 in the parity game E(ϕ(ϕ(x)), (T [x 7→ S], s)).
55
2.3. THE TRANSITIVE AND SYMMETRIC CASE
Let B := kϕ(x)kT [x7→S] . By construction of σ ∗ we have that for all vertices
of the form hx, vi which are reachable by σ ∗ it holds that v ∈ B. Then, by
applying Lemma 1.30, σ ∗ can be converted into a winning strategy for Player
0 in the evaluation game E(ϕ(ϕ(x), (T [x 7→ B], s)). By Theorem 1.28, we have
that
s ∈ kϕ(ϕ(B))kT
which can be reformulated as s ∈ kϕ(ϕ(ϕ(⊤)))kT or s ∈ kϕ(A)kT . Therefore,
we have proved Equation 2.1 and completed the proof.
Based on the previous result, we can therefore define a syntactical translation which associates to every µ-formula a modal formula and preserves logical
equivalence on transitive and symmetric transition systems.
t
Definition 2.8. The syntactical translation () : Lµ → LM is defined as the
identity for all propositional variables, ⊥ and ⊤, such that it distributes over
boolean and modal connectives, and such that
t
t
(µx.ϕ)t = wn(ϕ(ϕ(⊥)))
and (νx.ϕ)t = wn(ϕ(ϕ(⊤))) .
Note that (ϕ)t is defined via an application of ()t either to a strict subformula ψ of ϕ, or to a formula whose rank, by Lemma 1.21, is strictly smaller
than the rank of ϕ. Thus ()t terminates and is well-defined.
The next corollary proves that on transitive and symmetric models, the
st
. Its proof
semantical hierarchy of the µ-calculus collapses to the class ∆µT
1
goes by induction on the rank of a formula and uses Theorem 2.7.
Corollary 2.9. On transitive and symmetric transition systems we have that
kϕkT = kϕt kT .
Example 2.10. If we look at our example from Section 3.4, for “always eventually p”, we have that
st
kνx.(µy.(p ∨ 3y)) ∧ x)kT = k(p ∨ 3p) ∧ (p ∨ 3p)kT
st
and for “there is a path where p holds infinitely often”, we have that
st
kνx.µy.((p ∨ 3y)) ∧ 3x)kT
=
st
k p ∨ 3(p ∧ 3((p ∨ 3(p ∧ 3⊤)) ∧ 3⊤) ∧ 3((p ∨ 3(p ∧ 3⊤)) ∧ 3⊤)kT .
Remark 2.11. Because the previous proof applies to any S5 model, that is, for
every T ∈ Trst we have that:
kϕkT = kϕt kT
The fact that the modal µ-calculus hierarchy for S5-models collapses to the
pure modal fragment is indeed not surprising since for a S5-formula ϕ there
are only finitely many formulae with the same propositional variables which
are not equivalent over Trst and, therefore, it can easily be shown that for all
rst
rst
νx.ϕ(x) ∈ Lµ there is a n ∈ N such that kϕn (⊤)kT
= kνx.ϕkT . The
existence of only finitely many non equivalent formulae follows from the fact
that for all S5-formulae ϕ there is a conjunctive modal normal form ψ such that
ψ ≡ δ1 ∧ δ2 ∧ .. ∧ δn where δ ≡ α ∨ β1 ∨ β2 ∨ ...βn ∨ 3γ1 ∨ 3γ2 ∨ .. ∨ 3γm
and α, βi and γj are propositional formulae2 .
2 Cf.
Chapter 5 in [60].
56
CHAPTER 2.
2.4
The transitive case
We show that the modal µ-calculus hierarchy over Tt collapses to the alternationfree fragment. This is done in four parts starting from subsection two. First,
any modal µ-formula is reduced to a semantically equivalent formula τ (ϕ) such
that normalized strategies on evaluation games – which will be introduced in
the third subsection – have certain interesting properties. Then, we encode
such normalized winning strategies in modal µ-formulae and, finally, we show
the collapse for finite transitive transition system and, by using the previously
proved finite model theorem, generalize it to all transitive transition systems.
In the next subsection some technical notions like the one of unfolding a
formula in a model are introduced and some properties are proved.
2.4.1
Some technical preliminaries
Recall that we suppose all µ-formulae well-named. First we introduce the unfolding of a formula3 .
Definition 2.12. Let ϕ and ψ be µ-formulae such that {x1 , . . . , xn } = X ⊆
bound(ϕ). The unfolding of ψ in ϕ over X, unf X
ϕ (ψ), is the formula defined
∅
recursively such that unf ϕ (ψ) ≡ ψ and such that if X is of the form {x1 , . . . , xn }
then
X −1
X −n
unf X
(ϕx1 ), . . . , xn /unf ϕ
(ϕxn )]
ϕ (ψ) ≡ ψ[x1 /unf ϕ
where X −i = {x1 , . . . , xi−1 , xi+1 , . . . , xn }.
It can easily be seen that we have X ∩ free(unf X
ϕ (ψ)) = ∅.
In order to explain semantically the unfolding of a formula we introduce for
each transition system T the transition system induced by ϕ, T ϕ . For every
variable x ∈ bound(ϕ) we define a natural number l(x) recursively such that
l(x) = 0 if free(ϕx ) ∩ bound(ϕ) = ∅ and such that
l(x) = max{l(xi ) : xi ∈ free(ϕx ) ∩ bound(ϕ)} + 1
in the opposite case. For all transition systems T with valuation λ and for all
0 ≤ i ≤ max{l(x) : x ∈ bound(ϕ)} =: N we define new valuations λi and
transition systems T i such that λ0 = λ and T 0 = T , and such that T k+1 is
identical to T k except for the valuation λk+1 which is defined as follows:
• λk+1 |Prop\bound(ϕ) = λk |Prop\bound(ϕ) ;
• if x ∈ bound(ϕ):
k+1
λ
(x) =
(
λk (x)
kϕx kT k
if l(x) 6= k + 1
if l(x) = k + 1.
We define T ϕ to be T N and λϕ = λN . Note, that if we have a formula ψ such
that free(ψ) ∩ bound(ϕ) is empty then, since the denotation of ϕ is independent
of the valuation of the bound variables, we have kψkT = kψkT ϕ . In particular,
we have kϕkT = kϕkT ϕ . Moreover note that for all xi ∈ bound(ϕ) it holds that
λϕ (xi ) = kϕxi kT ϕ .
3 Since we restrict the unfolding of the free variables of a µ-formula to a certain fixed set of
propositional variables, this notion generalizes in some sense the one of closure of a formula,
introduced by Kozen in [73].
57
2.4. THE TRANSITIVE CASE
Lemma 2.13. For all formulae ϕ, all subformulae ψ ≤ ϕ, all X ⊆ bound(ϕ),
and all transition systems T we have that
kψkT ϕ = kunf X
ϕ (ψ)kT ϕ .
Proof. By induction on the size of X. If X is empty, then by definition of
unfolding we have that
unf X
ϕ (ψ) ≡ ψ
and the claim is trivial. For the inductive step, suppose that X ∩ free(ψ) is the
set {x1 , . . . , xm }. Hence, by definition we have
X
unf X
ϕ (ψ) ≡ ψ[x1 /unf ϕ
−1
X
(ϕx1 ), . . . , xm /unf ϕ
X
Since bound(ψ) ∩ free(ϕxi ) = ∅ and free(unf ϕ
X −1
bound(ψ) ∩ free((unf ϕ
(ϕxi ))
get that
and Lemma 1.19.1 we get
−1
−m
(ϕxm )].
(ϕxi )) ⊆ free(ϕxi ) for all i we
= ∅. Therefore, by induction hypothesis
kunf X
ϕ (ψ)kT ϕ = kψkT ϕ [x1 7→kϕx1 kT ϕ ,...,xm 7→kϕxm kT ϕ ] .
Since for all xi we have that λϕ (xi ) = kϕxi kT ϕ we get
kunf X
ϕ (ψ)kT ϕ = kψkT ϕ .
The previous lemma tells us that on the transition system induced by ϕ the
denotation of any subformula of ϕ and the denotation of any of its unfolding
over ϕ are the same.
Other usefull properties of T ϕ are summarized in the next lemma.
Lemma 2.14. Let T = (S, →T , λT ) be a transition system, ϕ any µ-formula and
ψ ≤ ϕ.Then:
(1) For every X ⊆ bound(ϕ) we have
kψ free(X) kT ϕ = kψkT ϕ .
(2) For every X1 , X2 ⊆ bound(ϕ), where X1 ∩ X2 = ∅, we have
2
kunf X
ψ free(X1 ) kT ϕ = kψkT ϕ .
ϕfree(X1 )
(3) For every X1 , X2 ⊆ bound(ϕ), where X1 ∩ X2 = ∅, we have
X2 free(X1 )
kunf ϕ
ψ
kT ϕ = kψkT ϕ .
ϕ
Proof. Part 1. By Lemma 1.19.2 and since kϕx kT ϕ = λT (x) for every variable
x ∈ bound(ϕ) we get
kψ free(X) kT ϕ = kψ free(X) [x1 /ϕx1 , . . . , xn /ϕxn ]kT ϕ .
The proof ends with a straightforward induction on the structure of ψ proving
that for all transition systems T we have
kψkT ϕ = kψ free(X) [x1 /ϕx1 , . . . , xn /ϕxn ]kT ϕ .
58
CHAPTER 2.
The only non trivial step is the one where ψ is of the form ηx.α (η ∈ {µ, ν}).
In this case, note that if any xi appears free in α then x appears only bounded
in ϕxi .
Part 2. We prove the equation by induction on the size of X2 . If X2 is
empty, the equation holds by the previous point. For the inductive step, given
{xi1 , . . . , xik } = X2 ∩free(ψ free(X1 ) ), we have that by definition of unf the formula
2
kunf X
ψ free(X1 ) kT ϕ is equal to
ϕfree(X1 )
X
−i1
X
−ik
2
2
(ϕfree(X1 ) )xi1 , . . . , xik /unf ϕfree(X
kψ free(X1 ) [xi1 /unf ϕfree(X
(ϕfree(X1 ) )xik ]kT ϕ .
1)
1)
X
−i1
2
Since free(unf ϕfree(X
(ϕfree(X1 ) )xi1 ) ⊆ free(ϕfree(X1 ) )xi1 ) and since we have that
1)
free(ϕfree(X1 ) )xi1 ) ∩ bound(ψ free(X1 ) ) = ∅ we get
X
−i1
2
free(unf ϕfree(X
(ϕfree(X1 ) )xi1 ) ∩ bound(ψ free(X1 ) ) = ∅.
1)
With Lemma 1.19.1 we get the equality with
kψ free(X1 ) k
T ϕ [xi1 7→kunf
−i
−i
X2 k
X2 1
(ϕfree(X1 ) )xi
(ϕfree(X1 ) )xi kT ϕ ,...,xik 7→kunf free(X
)
free(X
1
1)
1
k
ϕ
ϕ
kT ϕ ]
and by induction hypothesis this expression is equal to
kψ free(X1 ) kT ϕ [xi1 7→kϕxi
1
kT ϕ ,...,xik 7→kϕxi kT ϕ ] .
k
Since in T ϕ we have that λ(xij ) = kϕxij kT ϕ the last expression is equal to
kψ free(X1 ) kT ϕ .
Part 3. Suppose {xi1 , . . . , xik } = X2 ∩ free(ψ free(X1 ) ). Following the same
free(X1 )
2
argumentation as in part 2 we get that kunf X
kT ϕ is equal to
ϕ ψ
kψ free(X1 ) k
X
−i1
T ϕ [xi1 7→kunf ϕ 2
X
1
.
−ik
ϕxi kT ϕ ,...,xik 7→kunf ϕ 2
ϕ x i kT ϕ ]
k
With Lemma 2.13 we get the equality with
kψ free(X1 ) kT ϕ [xi1 7→kϕxi
1
kT ϕ ,...,xik 7→kϕxi kT ϕ ] ,
k
ϕ
and because in T we have that λ(xij ) = kϕxij kT ϕ the last expression is equal
to kψ free(X1 ) kT ϕ which by part 1 is equal to kψkT ϕ .
Lemma 2.15. Let ϕ be a µ-formula and T = (S, →T , λT ) be a transition system.
For all X ⊆ bound(ϕ), all xk ∈ X = bound(ϕ) \ X, all ψ ≤ ϕ and all x ∈
/ X we
have that
ψ −X kT ⊆ kunf
(1) kunf X
ϕ−X
X∪{xk }
ϕ−X
−k
ψ −X
−k
(2) kunf X
ψ −X kT ⊆ kunf bound(ϕ)
ψkT ,
ϕ
ϕ−X
(ϕ−X )x kT ϕ ⊆ kϕx kT ϕ ,
(3) kunf X
ϕ−X
(4) kψ −X kT ϕ−X ⊆ kψkT ϕ .
kT ,
59
2.4. THE TRANSITIVE CASE
ψ −X , (T , s)).
Proof. Suppose σ is a winning strategy for Player 0 in E(unf X
ϕ−X
ψ −X , si and
By definition, any winning play for Player 0 starting from hunf X
ϕ−X
compatible with σ do not reach a position of type h⊥, s′ i. Thus, this strategy
−k
X∪{x }
determines a winning strategy for Player 0 in E(unf −X −kk ψ −X , (T , s)). Part 1
ϕ
is then obtained by applying Theorem 1.28. Part 2 follows by a finite reiteration
of part 1. In order to obtain part 3 just apply Lemma 2.13 to part 2 and note
that, since x ∈
/ X, (ϕ−X )x ≡ (ϕx )−X . Part 4 is also a consequence of an
application of Lemma 2.13 to part 2.
2.4.2
A first reduction
We begin with a lemma whose proof is standard.
Lemma 2.16. Let T be a transitive transition system and let s, s′ be two states
such that s →T s′ . For all µ-formulae ϕ we have that
s ∈ kϕkT
s′ ∈ kϕkT
=⇒
s′ ∈ k3ϕkT
=⇒
and
s ∈ k3ϕkT .
The next theorem assures us that over transitive transition systems, a greatest fixpoint can be eliminated when the bounded variable is guarded by an
universal modality.
Theorem 2.17. Let T be a transitive transition system and let νx.ϕ(x) be a
well-named formula such that x is weakly universal. We have that
kνx.ϕ(x)kT = kϕ(ϕ(⊤))kT .
Proof. The ⊆ inclusion is clear. For the ⊇ inclusion, define A = kϕ(ϕ(⊤))kT ;
by definition of greatest fixpoint it is enough to show that we have
A ⊆ kϕ(A)kT .
(2.2)
First, recall that we assume that νx.ϕ(x) is well-named. This means that in the
formula ϕ(x) the variable x is in the scope of a modal operator and, therefore,
we can assume that ϕ is of the form β(α(x)). Moreover x occurs only once in
ϕ. This implies that α(x) and α(ϕ(x)) occur only once in the formula tree
of ϕ(ϕ(x)). Let s ∈ A, by Theorem 1.28 there is a winning strategy σ for Player
0 in the evaluation game E(ϕ(ϕ(x)), (T [x 7→ S], s)). Let π be an arbitrary play
consistent with σ. If π reaches a vertex of the form hα(x), s′ i then the same
play passes a vertex of the form hα(ϕ(x)), s′′ i, with α(x) ≤ α(ϕ(x)) and
s′ reachable from s′′ in T [x 7→ S]. Since σ is a winning strategy for Player 0 by
Proposition 1.33 we have that
s′′ ∈ kα(ϕ(x))kT [x7→S]
and s′ ∈ kα(x)kT [x7→S] .
Since T [x 7→ S] is transitive we have that s′′ →T [x7→S] s′ and, by applying
Lemma 2.16, we have
s′ ∈ kα(ϕ(x))kT [x7→S] .
Hence, we have shown that for all plays π consistent with σ, if π reaches a vertex
of the form hα(x), s′ i then, by Theorem 1.28, there is a winning strategy for
60
CHAPTER 2.
Player 0 in the evaluation game E(α(ϕ(x)), (T [x 7→ S], s′ )). A fortiori, this
implies that if π reaches a vertex of the form hα(x), s′ i then there is a winning
strategy σhα(x),s′ i for Player 0 in E(α(x), (T [x 7→ kϕ(x)kT [x7→S] ], s′ )). Therefore, since kϕ(x)kT [x7→S] ⊆ S, the strategy σ ∗ given by following σ but switching to the corresponding σhα(x),s′ i when a position of the form hα(x), s′ i is
reached, is winning for Player 0 in the parity game E(ϕ(ϕ(x)), (T [x 7→ S], s)).
Let B := kϕ(x)kT [x7→S] . By construction of σ ∗ we have that for all vertices
of the form hx, vi which are reachable by σ ∗ it holds that v ∈ B. Then, by
applying Lemma 1.30, σ ∗ can be converted into a winning strategy for Player 0
in E(ϕ(ϕ(x), (T [x 7→ B], s)). By Theorem 1.28, we have that
s ∈ kϕ(ϕ(B))kT [x7→S]
which can be reformulated as s ∈ kϕ(ϕ(ϕ(⊤)))kT or s ∈ kϕ(A)kT . Therefore,
we have proved Equation 2.2 and completed the proof.
Definition 2.18. The syntactical translation τ : Lµ → Lµ is defined recursively
on the structure of the formula such that τ (p) = p, τ (¬p) = ¬p, τ (⊥) = ⊥ and
τ (⊤) = ⊤, such that it distributes over boolean and modal connectives, and such
that
• τ (µx.ϕ) = τ wn(ϕ(ϕ(⊥))) , x is weakly existential in ϕ
• τ (µx.ϕ) = wn(µx.τ (ϕ)), x is universal in ϕ
• τ (νx.ϕ) = τ wn(ϕ(ϕ(⊤))) , x is weakly universal in ϕ
• τ (νx.ϕ) = wn(νx.τ (ϕ)), x is existential in ϕ.
First, note that in each defining clause τ (ϕ) is defined via an application of
τ to a formula whose rank, by Lemma 1.21, is strictly smaller than the rank
of ϕ. Thus τ terminates and is well-defined. Note also, that it can be proved
by induction on the structure of ϕ that all variables which are existential (resp.
universal) in ϕ are weakly existential (resp. universal) in τ (ϕ) and that therefore
for all µx.α ≤ τ (ϕ) we have that x is weakly universal and for all νx.α ≤ τ (ϕ)
we have that x is weakly existential.
Corollary 2.19. On transitive transition systems we have that
kϕkT = kτ (ϕ)kT .
Proof. By induction on rank(ϕ). If rank(ϕ) = 1 or rank(ϕ) is a successor ordinal
the proof is straightforward. If rank(ϕ) is a limit ordinal then ϕ is of the form
ηx.α. We distinguish four cases. If ϕ is of the form νx.α and x is existential in
ϕ the induction step is straightforward. Similarly for ϕ of the form µx.α and
x is universal in ϕ. If ϕ is of the form νx.α and x is in the scope of a in ϕ
the induction step follows from Theorem 2.17 and Lemma 1.20. In the third
case, if ϕ is of the form µx.α and x is in the scope of a 3 in ϕ then ¬ϕ is of
the form νx.¬α[x/¬x] and x is in the scope of a in ¬ϕ. Since in this case
rank(ϕ) = rank(¬ϕ) we can apply the induction step as in the third case.
61
2.4. THE TRANSITIVE CASE
2.4.3
Normalizing the winning strategies
Let T be a transitive transition system and ϕ a µ-formula. Consider an arbitrary
(memoryless) strategy σ for Player 0, not necessarily winning. We define the
restriction of E(ϕ, (T , s0 )) on σ, denoted by E|σ (ϕ, (T , s0 )), as follows:
• The set of positions V |σ of the restriction is given by all nodes which
are the positions of some play compatible with σ starting from position
hϕ, s0 i,
• The arena of E|σ (ϕ, (T , s0 )) is the triple hV0 |σ , V1 |σ , E|σ i where:
(1)
(2)
(3)
(4)
V0 |σ = ∅,
V1 |σ = V |σ ,
if hψ, si ∈ V |σ ∩ V1 then E|σ (hψ, si) = E(hψ, si), and
if hψ, si ∈ V |σ ∩ V0 then E|σ (hψ, si) = {σ(hψ, si)}.
• The ranking function Ω|σ is given by the restriction of Ω on V |σ .
Note, that if T is finite then V |σ is finite, too. We have that in E|σ (ϕ, (T , s0 ))
the only Player who can move is Player 1. This can be done because the moves
for Player 0 are already completely determined by the (memoryless) strategy σ.
Clearly, any play in E|σ (ϕ, (T , s0 )) is a play in E(ϕ, (T , s0 )) compatible with σ.
We say that a play π in E|σ (ϕ, (T , s0 )) is winning for Player 0 if and only if the
play π is winning for Player 0 in E(ϕ, (T , s0 )). If σ is a winning strategy for
Player 0 then any play in E|σ (ϕ, (T , s0 )) is winning for Player 0.
Example 2.20. Consider the arena of Example 1.29 depicted in Figure 1.5. The
non-dotted part of the picture represents the arena of a restricted evaluation
game.
We now formally define a measure, which roughly associates to every node of
a restricted evaluation game the height of the corresponding strongly connected
component in the scc tree.
Definition 2.21. Let T be a finite transitive transition system and ϕ a µformula. Suppose there is a winning strategy σ for Player 0 in the parity game
E(ϕ, (T , s0 )). Then, for every position hψ, si of E|σ (ϕ, (T , s0 )), we define a
measure d(hψ, si). We distinguish two cases in the definition, depending on
whether the strongly connected component scc(hψ, si) of hψ, si in E|σ (ϕ, (T , s0 ))
is empty or not:
(1) scc(hψ, si) = ∅ :
(
0
d(hψ, si) =
max{d(hφ, s′ i) : hφ, s′ i ∈ E|σ (hψ, si)} + 1
if E|σ (hψ, si) = ∅
else
(2) scc(ψ, s) 6= ∅ :
d(hψ, si) = 0
else
if
[
{E|σ (hα, si) : hα, si ∈ scc(ψ, s)} \ scc(ψ, s) = ∅,
d(hψ, si) = max{d(hφ, s′ i) : hφ, s′ i ∈
/ scc(hψ, si) and exists
hξ, s′′ i ∈ scc(hψ, si) with hφ, s′ i ∈ E|σ (hξ, s′′ i)} + 1.
62
CHAPTER 2.
For all finite transition systems d is a well-defined measure. Indeed, if we
have a finite transition system we obviously have a finite arena which can be
collapsed to a finite and well-founded graph by identifying all vertices in the
arena which are in the same strongly connected component. It is clear that on
finite and well-founded graphs d is well-defined. By noticing that on the original
arena the measure of a vertex corresponds to its measure of the collapsed arena
we get that d is well-defined.
Lemma 2.22. Let T be a finite transitive transition system and ϕ ∈ Σµ2 . Suppose
there is a winning strategy σ for Player 0 in the parity game E(ϕ, (T , s0 )). If
y ∈ bound(ϕ) is a µ-variable, then for every position hy, si ∈ V |σ , we have that
scc(hy, si) = ∅.
Proof. If scc(hy, si) 6= ∅ then Player 1 can find a play π in the restricted game
E|σ (ϕ, (T , s0 )) where hy, si occurs infinitely often, since in E|σ (ϕ, (T , s0 )) only
Player 1 moves and therefore can stay as long as he wants in a strongly connected
component. Recall that ϕ ∈ Σµ2 . Thus, there is no ν-variable free in ϕy .
Moreover, if y ∈ free(ϕx ), where x is an arbitrary ν-variable, we have that
Ω(hy, si) is strictly greater than the priorities of x and ϕx positions. Therefore,
π is winning for Player 1. But since π is compatible with σ, the play must be
winning for Player 0, too. A contradiction.
Given the restriction of E(ϕ, (T , s0 )) on a winning strategy σ and the measure d on V |σ we define the normalization of σ, denoted by σ N , as follows:
• For all positions h3β, s′ i ∈ V |σ we have that
σ N (h3β, s′ i) = σ(h3β, s′ i),
if d(σ(h3β, s′ i)) is the minimum of the set {d(hβ, s̄i) : hβ, s̄i ∈ E(h3β, s′ i)},
where any hβ, s̄i ∈ E(h3β, s′ i) has to be reachable from h3β, s′ i in V |σ .
Else
σ N (h3β, s′ i) = hβ, s′′ i,
where hβ, s′′ i ∈ E(h3β, s′ i) is a vertex reachable from h3β, s′ i in V |σ such
that d(hβ, s′′ i) is the minimum of the set {d(hβ, s̄i) : hβ, s̄i ∈ E(h3β, s′ i)}
where any hβ, s̄i ∈ E(h3β, s′ i) has to be reachable from h3β, s′ i in V |σ .
• If ψ is not of the form 3β then we simply set σ N (hψ, si) = σ(hψ, si).
Intuitively, given a winning strategy σ for Player 0 on E(ϕ, (T , s0 )), the normalized strategy σ N for Player 0 is given by adapting σ such that for all vertexes
of the form h3β, s′ i Player 0 moves to a vertex hβ, s′′ i whose measure is the
minimal measure of all positions of the type hβ, s̄i reachable from h3β, s′ i which
are still winning in E(ϕ, (T , s0 )). We have the following lemma.
Lemma 2.23. Let T be a finite transitive transition system. If σ is a winning
strategy for Player 0 on E(ϕ, (T , s0 )) then σ N is a winning strategy for Player
0 on E(ϕ, (T , s0 )), too.
Proof. First we prove the following claim:
Claim : E|σ and E|σN coincide on every non empty scc of E|σN (ϕ, (T , s0 )).
The proof of the claim goes as follows. If there is no position of the form
h3β, si in a scc of E|σN (ϕ, (T , s0 )), the claim is trivially verified. Consider now
2.4. THE TRANSITIVE CASE
63
an arbitrary scc(h3β, si) of E|σN (ϕ, (T , s0 )). Let hψ, ti ∈ scc(h3β, si), in order
to prove the claim we have to show that E|σN (hψ, ti) = E|σ (hψ, ti).
(a) If ψ is not of the form 3α, then E|σN (hψ, ti) = E|σ (hψ, ti).
(b) For the case where ψ = 3α then suppose that E|σN (hψ, ti) 6= E|σ (hψ, ti) and
that E|σN (hψ, ti) = {hα, t′ i}. Note, that by construction of σ N the position
hα, t′ i is the only successor of hψ, ti. Since E|σN (hψ, ti) 6= E|σ (hψ, ti) it must
hold that
d(hα, t′ i) < d(hψ, ti)
(2.3)
where d is the depth defined on E|σ (ϕ, (T , s0 )). Since scc(h3α, ti) 6= ∅ and
since hα, t′ i is the only position reachable in one step from h3α, ti we have
that hα, t′ i ∈ scc(h3α, ti) and therefore that h3α, ti is reachable from hα, t′ i
in E|σN (ϕ, (T , s0 )). Since reachability in E|σN (ϕ, (T , s0 )) implies reachability
in E|σ (ϕ, (T , s0 )) we can infer that d(hα, t′ i) ≥ d(h3α, ti), where d is the
depth defined on E|σ (ϕ, (T , s0 )). This is a contradiction to point 2.3 and
therefore the claim is proved.
Consider an arbitrary play π in the graph of E|σN (ϕ, (T , s0 )). If π is finite, then
by construction of the normalized arena the play is winning for Player 0. If π is
infinite then from a certain position, say hα, ti, we are in a scc of E|σN (ϕ, (T , s0 )).
But then by the previous claim after hα, ti the strategies of σ and σ N coincide.
Since by construction of σ N the position hα, ti is winning in E|σ (ϕ, (T , s0 )) the
highest priority appearing infinitely often in π must be even and, therefore π is
a winning play in E(ϕ, (T , s0 )) for Player 0.
In the next lemma we prove that, when considering Σµ2 -formulae, normalized
strategies have a nice and very usefull property.
Lemma 2.24. Let T be a finite transitive transition system and ϕ ∈ Σµ2 such that
all ν-variables are weakly existential. Let σ N be a normalized winning strategy for
Player 0 on E(ϕ, (T , s0 )). Consider a position hx, s1 i in E|σN (ϕ, (T , s0 )) where
x ∈ bound(ϕ) is a ν-variable. Then, if there is a position hy, s2 i reachable from
hx, s1 i in V |σN , where y ∈ bound(ϕ) is a µ-variable, then there is no position
hx, s3 i reachable from hy, s2 i in V |σN .
Proof. Suppose there is a play π consistent with σ N such that we have the
following regenerations: hx, s1 i then hy, s2 i and then hx, s3 i, where x is a νvariable and y a µ-variable. Note that, since ϕ ∈ Σµ2 , we have that y ∈ free(ϕx ),
and therefore ϕx < ϕy . This implies that in π we must have positions of the
form h3(β(x)), s′1 i and hβ(x), s′′1 i before hx, s1 i, and also positions of the form
h3(β(x)), s′3 i and hβ(x), s′′3 i before hx, s3 i but after hy, s2 i. By construction of
normalized strategy and by the transitivity of the transition system T it holds
that d(hβ(x), s′′1 i) = d(hβ(x), s′′3 i) but also that d(hβ(x), s′′1 i) = d(hβ(x), s′′3 i) =
d(hy, s2 i). This implies that scc(hy, s2 i) 6= ∅. Because σ N is a winning strategy
for Player 0, by Lemma 2.22 we get the desired contradiction.
We immediately can restate the previous lemma as the following theorem.
Theorem 2.25. Suppose a finite transitive transition system T , a formula ϕ ∈
Σµ2 such that all ν-variables are weakly existential and a normalized winning
strategy, σ N , of Player 0 in E(ϕ, (T , s)). If in a play π consistent with σ N there
64
CHAPTER 2.
is a regeneration of a ν-variable x then either there is no more regeneration of
a µ-variable after the first regeneration of x or, if there is such a regeneration
of a µ-variable, then after this position there is no more regeneration of x.
2.4.4
Encoding normalized winning strategies
In the last theorem of the previous subsection, we essentially verified that given
a finite transitive transition system T , a formula ϕ ∈ Σµ2 such that all νvariables are weakly existential and a normalized winning strategy of Player
0 in E(ϕ, (T , s)), every play consistent with this strategy does not have any “alternation” between positions with a least fixpoint formula and position with a
greatest fixpoint formula. This fact suggests the idea that, if we are able to construct a formula ψ based on the original formula ϕ and on the general behavior
of normalized winning strategies, on transitive models ψ should be proven to be
equivalent to the original µ-formula and to be alternation free.
This is the approach we pursue in the present and next subsections. More
+
′
′
precisely, in Definition 2.26 we define the formulae NS−
ϕ (X , y) and NSϕ (x, X )
that will be used to encode the main properties of the normalization of winning
strategies of ϕ given by Theorem 2.25. Encoding, in this context, will be formalized in the two main Lemmas of the section, Lemmas 2.28 and 2.29. The
intuition behind these formulae is the following:
′
• NS−
ϕ (X , y) reflects the fact that we are regenerating y and any ν-variable
regenerated afterwards will be an element of X ′ ,
′
• NS+
ϕ (x, X ) reflects the fact that we are regenerating x and if we regenerate any µ-variable then afterwards any ν-variable regenerated will be an
element of X ′ .
Then in the next subsection we show how to construct from a given formula
′
ϕ ∈ Σµ2 an equivalent alternation free formula based on the formulae NS−
ϕ (X , y)
+
and NSϕ (x, X ′ )
In the sequel, in order to ease notation, we write a formula of the form
free(X)
ϕy
instead of (ϕfree(X) )y .
Definition 2.26. Let ϕ be a Σµ2 -formula. Let Y = {y1 , . . . , yk } be the set of
all µ-variables in ϕ and X be the set of all ν-variables in ϕ. For all subsets of
X ′ ⊂ X, all ν-variables x such that x ∈ X/X ′ and all µ-variables y we define
−
′
′
′
the formulae NS+
ϕ (x, X ) and NSϕ (X , y) recursively on the size of X such that
Y
−X
NS−
)y )
ϕ (∅, y) ≡ unf ϕ−X ((ϕ
and, such that
X
−
free(Y )
NS+
)[y1 /NS−
ϕ (x, ∅) ≡ (unf ϕfree(Y ) ϕx
ϕ (∅, y1 ), . . . , yk /NSϕ (∅, yk )].
If X ′ = {xi1 , . . . , xil } and X ′ = X \ X ′ , then
free(X ′ )
Y
−X
′
)y
NS−
ϕ (X , y) ≡ (unf (ϕ−X ′ )free(X ′ ) (ϕ
′
)[
−i
xi1 /NS+−X ′ (xi1 , X ′ 1 ),
ϕ
..
.
xik /NS+−X ′ (xil , X ′ −il )],
ϕ
65
2.4. THE TRANSITIVE CASE
and
′
NS+
ϕ (x, X ) ≡
′
free(Y ∪X ′ )
(unf X
ϕfree(Y ∪X ′ ) ϕx
)[
′
y1 /NS−
ϕ (X , y1 ),
..
.
′
yk /NS−
ϕ (X , yk ),
−i
+
xi1 /NSϕ (xi1 , X ′ 1 ),
..
.
′
xil /NS+
ϕ (xil , X
−il
)].
Note that by construction we have that for every ν-variable x, every µ−
′
′
variable y and every set of ν-variables X ′ , free(NS+
ϕ (x, X )), free(NSϕ (X , y)) ⊆
+
−
′
′
free(ϕ) and bound(NSϕ (x, X )), bound(NSϕ (X , y)) ⊆ bound(ϕ).
Lemma 2.27. Let ϕ ∈ Σµ2 , y be µ-variable in ϕ and X ′ be a proper subset of the
set of all ν-variables. Suppose that xi is a ν-variable such that xi 6∈ X ′ . We
have that
µ
+
′
′
NS−
ϕ (X , y), NSϕ (xi , X ) ∈ ∆2 .
Proof. The proof goes by induction on the size of X ′ . If X ′ = ∅ then clearly
µ
µ
+
′
′
NS−
ϕ (X , Y ) ∈ Σ1 and, by definition of the formula, NSϕ (x, X ) ∈ ∆2 . The
µ
induction step follows from the definitions by noting that the class ∆2 is closed
under substitution of ∆µ2 formulae if no new variable is bound.
Lemma 2.28. Let ϕ be a Σµ2 -formula and X be the set of all ν-variables in ϕ.
Suppose that all x ∈ X are weakly existential. Let (T , s0 ) be a finite transitive
transition system such that there is a normalized winning strategy σ N in the
evaluation game E(ϕ, (T , s0 )). The following holds for every X ′ ⊆ X where
X ′ = X/X ′:
(1) If there is a play consistent with σ N which reaches a position hy, si (y a µvariable in ϕ) such that on this play before hy, si there are positions hx, si
for all x ∈ X ′ then it holds that
′
s ∈ kNS−
ϕ (X , y)kT ϕ .
(2) If there is a play consistent with σ N which reaches for the first time a
position hx, si (x a ν-variable in ϕ) such that on this play before hx, si
there are positions hx, si for all x ∈ X ′ \ {x} then it holds that
′
s ∈ kNS+
ϕ (x, X )kT ϕ .
Proof. Let Y = {y1 , . . . , yk } be the set of all µ-variables in ϕ. We prove the two
points simultaneously by induction on the size of X ′ . If X ′ = ∅ we have that
Y
−X
)y ). If there is a play consistent with σ N reaching a
NS−
ϕ (∅, y) ≡ unf ϕ−X ((ϕ
position of the form hy, si whereby for all ν-variables there has been a regeneration in this play before, then, since σ N is a normalized strategy, by Theorem
2.25 there can not be any regeneration of a ν-variable after hy, si. Therefore σ N
determines a winning strategy in
E(unf Yϕ−X ((ϕ−X )y ), (T , s))
66
CHAPTER 2.
and with Theorem 1.28 we get the induction base for part 1. For part 2 recall
that
X
−
free(Y )
NS+
)[y1 /NS−
ϕ (x, ∅) ≡ (unf ϕfree(Y ) ϕx
ϕ (∅, y1 ), . . . , yk /NSϕ (∅, yk )].
Suppose that there is a play consistent with σ N which reaches for the first time
a position hx, si (x ∈ X) such that on this play before hx, si there are positions
hx, si for all x ∈ X \ {x}. Then, since σ N is a normalized strategy, by Theorem
2.25 for every play extending this position which is compatible with σ N , either
there are only regenerations of ν-variables, or, if there is a regeneration of a
hy, si, then after this regeneration there is no more regeneration of a ν-variable.
Therefore σ N determines a winning strategy in
−
free(Y )
E((unf X
)[y1 /NS−
ϕ (∅, y1 ), . . . , yk /NSϕ (∅, yk )], (T , s))
ϕfree(Y ) ϕx
and with Theorem 1.28 we get the induction base for part 2.
For the induction step of part 1, let X ′ = {xi1 , . . . , xil } and let hy, si be a
position of a play consistent with σ N such that all x ∈ X ′ have been regenerated
before. Then, by Theorem 2.25 for all ν-variables xi regenerated afterwards in
the play we have xi ∈ X ′ . By construction for such a position hxi , si i we will
have that all ν-variables in X ′ are regenerated before this position. Define
−i
X ′ = X ′ \ {xi }. It can easily be seen that hxi , si i satisfy the condition of
−i
part 2 and, since xi ∈ X ′ , that X ′ ( X ′ . Therefore, we can apply induction
hypothesis of part 2 and get
−i
si ∈ kNS+ (xi , X ′ )kT .
Recapitulating, we have that for all plays consistent with σ N starting from hy, si
−i
if a ν-variable xi is regenerated by a position hxi , si i then si ∈ kNS+ (xi , X ′ )kT
and otherwise we have only regenerations of µ-variables. But by Lemmas 1.19.1
and 1.30 this means that σ N gives us a winning strategy in the evaluation game
E(γ, (T , s))
where
γ≡
′
free(X ′ )
unf Y(ϕ−X ′ )free(X ′ ) ((ϕ−X )y
−i
)[ xi1 /NS+−X ′ (xi1 , X ′ 1 ),
ϕ
..
.
xil /NS+−X ′ (xil , X ′ −il )].
ϕ
′
NS−
ϕ (X , y)
By noting that γ ≡
and using Theorem 1.28 we finish the induction
step for part 1.
For the induction step of part 2 let hx, si be a position of a play consistent
with σ N such that all x ∈ X ′ have been regenerated before. There are only three
disjoint classes of winning plays (consistent with σ N ) extending the position
hx, si and they are obtained by considering all possible regenerations of bound
variables after this position:
(1) The class of plays in which afterwards we regenerate a xi ∈ X ′ in a position
hxi , si i, and before this position there was no regeneration of a µ-variable.
In this case we can apply the induction hypothesis for part 2 to the set
−i
X ′ and get
′ −i
si ∈ kNS+
)kT .
ϕ (xi , X
67
2.4. THE TRANSITIVE CASE
(2) The class of plays in which afterwards we regenerate a µ-variable y in a
position hy, sy i, and before this position there was no regeneration of a
xi ∈ X ′ . In this case, we can apply part 1, where the induction step is
already done, and get
′
sy ∈ kNS−
ϕ (X , y)kT .
(3) The class of plays in which there is no regeneration of z ∈ X ′ ∪ Y , but
there are eventually only regenerations of xi ∈ X ′ . Because these plays
are consistent with σ N , they are winning. Therefore, they are winning in
′
−(Y ∪X ′ )
the evaluation game E((unf X
), (T , s)), too.
ϕ−(Y ∪X ′ ) ϕx
By Lemmas 1.19.1 and 1.30 we have that
free(Y ∪X ′ )
′
s ∈ k(unf X
ϕfree(Y ∪X ′ ) ϕx
′
)[ y1 /NS−
ϕ (X , y1 ),
..
.
′
yk /NS−
ϕ (X , yk ),
′ −i1
xi1 /NS+
),
ϕ (xi1 , X
..
.
′
xil /NS+
ϕ (xil , X
−il
)]kT .
and this ends the induction step of part 2 and the proof.
Lemma 2.29. Let ϕ be a Σµ2 -formula and X be the set of all ν-variables in
ϕ. Suppose that all ν-variables are weakly existential. Then, for every finite
transitive transition system T and for every X ′ ⊆ X it holds that
(1) For every y ∈ Y we have
′
kNS−
ϕ (X , y)kT ϕ ⊆ kϕy kT ϕ , and
(2) for every x ∈ X ′ =: X/X ′ we have
′
kNS+
ϕ (x, X )kT ϕ ⊆ kϕx kT ϕ .
Proof. Let Y = {y1 , . . . , yk } be the set of all µ-variables. We prove the two
points simultaneously by induction on the size of X ′ . Suppose X ′ is empty.
Y
−X
Then we have that NS−
)y ) and by Lemma 2.15.3 we
ϕ (∅, y) ≡ unf ϕ−X ((ϕ
obtain
kunf Yϕ−X ((ϕ−X )y )kT ϕ ⊆ kϕy kT ϕ .
Therefore we complete the base case of the induction for part 1. For part 2
recall that
X
−
free(Y )
NS+
)[y1 /NS−
ϕ (x, ∅) ≡ (unf ϕfree(Y ) ϕx
ϕ (∅, y1 ), . . . , yk /NSϕ (∅, yk )].
Thus, by the induction base of part 1 and by Lemma 1.19.4, we have that
free(Y )
−
)[y1 /NS−
ϕ (∅, y1 ), . . . , yk /NSϕ (∅, yk )]kT ϕ
⊆
free(Y )
X
)[y1 /ϕy1 , . . . , ym /ϕym ]kT ϕ .
k(unf ϕfree(Y ) ϕx
k(unf X
ϕfree(Y ) ϕx
68
CHAPTER 2.
But because in T ϕ we have that λ(y) = kϕy kT ϕ and by applying Lemma 1.19.1
and Lemma 2.14.2, it holds that
free(Y )
k(unf X
)[y1 /ϕy1 , . . . , ym /ϕym ]kT ϕ ⊆ kϕx kT ϕ .
ϕfree(Y ) ϕx
Therefore
−
free(Y )
k(unf X
)[y1 /NS−
ϕ (∅, y1 ), . . . , yk /NSϕ (∅, yk )]kT ϕ ⊆ kϕx kT ϕ .
ϕfree(Y ) ϕx
This ends the induction base for both parts 1 and 2.
Let X ′ = {xi1 , . . . , xil }. For the induction step of part 1, recall that
′
−i
Y
′
−X free(X )
)
)y [ xi1 /NS+−X ′ (xi1 , X ′ 1 ),
NS−
ϕ (X , y) ≡ unf (ϕ−X ′ )free(X ′ ) ((ϕ
ϕ
..
.
−i
xil /NS+−X ′ (xil , X ′ l )].
′
ϕ
−X ′
By induction hypothesis, by Lemma 1.19.4 and because in T ϕ
the evaluation
′
of a variable xij ∈ X ′ is equal to k(ϕ−X )xij k ϕ−X ′ , we obtain
T
free(X ′ )
Y
′
−X
kNS−
)y
ϕ (X , y)kT ϕ = k unf (ϕ−X ′ )free(X ′ ) ((ϕ
′
)
−i
[xi1 /NS+−X ′ (xi1 , X ′ 1 ),
ϕ
..
.
−i
xil /NS+−X ′ (xil , X ′ l )]kT
ϕ
′
′
⊆ kunf Y(ϕ−X ′ )free(X ′ ) ((ϕ−X )free(X ) )y k
T ϕ−X
′
.
With Lemma 2.14.2 we obtain
′
′
kunf Y(ϕ−X ′ )free(X ′ ) ((ϕ−X )yfree(X ) )k
′
T ϕ−X
′
= k(ϕ−X )y k
T ϕ−X
′
.
′
Finally, because by Lemma 2.15.4 it holds that k(ϕ−X )y k ϕ−X ′ ⊆ kϕy kT ϕ we
T
get
′
kNS−
ϕ (X , y)kT ϕ ⊆ kϕy kT ϕ .
For the induction step of part 2 if X = X \ X ′ , then by induction hypothesis
and by part 1 we have for every finite transitive transition system T
′
kNS−
ϕ (X , y1 )kT ϕ
′
kNS−
ϕ (X , yk )]kT ϕ
−i
+
kNSϕ (xi1 X ′ 1 )kT ϕ
′
kNS+
ϕ (xil X
−il
)kT ϕ
⊆
..
.
⊆
⊆
..
.
⊆
kϕy1 kT ϕ ,
kϕyk kT ϕ ,
kϕxi1 kT ϕ ,
kϕxil kT ϕ .
Therefore, by Lemmas 1.19.4 and 1.19.1, and because for every z ∈ bound(ϕ)
69
2.4. THE TRANSITIVE CASE
we have that λ(z) = kϕz kT ϕ , we get
free(Y ∪X ′ )
′
k(unf X
ϕfree(Y ∪X ′ ) (ϕx
⊆
′
y1 /NS−
ϕ (X , y1 ),
..
.
′
yk /NS−
ϕ (X , yk )
−i
+
xi1 /NSϕ (xi1 , X ′ 1 ),
..
.
))[
′
xil /NS+
ϕ (xil , X
′
−il
)]kT ϕ
′
free(Y ∪X )
)kT ϕ .
kunf X
ϕfree(Y ∪X ′ ) (ϕx
Thus, we can apply Lemma 2.14.2 and obtain
′
′
free(Y ∪X )
k(unf X
))kT ϕ ⊆ kϕx kT ϕ .
ϕfree(Y ∪X ′ ) (ϕx
Because this implies that
′
kNS+
ϕ (x, X )kT ϕ ⊆ kϕx kT ϕ
this ends the induction step of part 2 and the proof of the Lemma.
2.4.5
The collapse over transitive models
Everything now is ready to prove the collapse of the µ-hierarchy over finite
transitive transition systems.
Definition 2.30. For the formula ϕ ∈ Σµ2 such that X = {x1 , . . . , xm } is the
set of all ν-variables in ϕ. We define a new formula ρ(ϕ) ∈ ∆µ2 such that
−1
−m
ρ(ϕ) ≡ ϕfree(X) [x1 /NS+
), . . . , xm /NS+
)].
ϕ (x1 , X
ϕ (xm , X
Remark 2.31. By Lemma 2.27 it can easily be seen that ρ(ϕ) is indeed a ∆µ2 formula.
Theorem 2.32. For all ϕ ∈ Σµ2 and all finite transitive transition systems T we
have that
kϕkT = kρ(τ (ϕ))kT .
Proof. First, we observe that τ (ϕ) ∈ Σµ2 and that by Corollary 2.19 we have
that kϕkT = kτ (ϕ)kT . Thus, we can assume that each ν-variable in ϕ ∈ Σµ2 is
weakly existential and any µ-variable weakly universal. If X = {x1 , . . . , xm } is
the set of all ν-variables in ϕ, by definition of ρ we have to prove that
−1
−m
kϕkT = kϕfree(X) [x1 /NS+
), . . . , xm /NS+
)]kT .
ϕ (x1 , X
ϕ (xm , X
−1
−m
“⊇”: Note that T [x1 7→ kNS+
)kT , . . . , xm 7→ kNS+
)kT k]
ϕ (x1 , X
ϕ (xm , X
+
+
−m
−1
ϕ
and T [x1 7→ kNSϕ (x1 , X )kT ϕ , . . . , xm 7→ kNSϕ (xm , X )kT ϕ k] agree on
+
−i
−i
the free variables of ϕfree(X) because kNS+
ϕ (xi , X )kT and kNSϕ (xi , X )kT ϕ
coincide for every xi ∈ X. Therefore we have that
−m
−1
)]kT =
), . . . , xm /NS+
kϕfree(X) [x1 /NS+
ϕ (xm , X
ϕ (x1 , X
+
+
−1
free(X)
kϕ
[x1 /NSϕ (x1 , X ), . . . , xm /NSϕ (xm , X −m )]kT ϕ
70
CHAPTER 2.
With Lemma 2.29 and, because all ν-variables appear positively in ϕ, by applying Lemma 1.19.4 we get that
−1
−m
kϕfree(X) [x1 /NS+
), . . . , xm /NS+
)]kT ϕ
ϕ (x1 , X
ϕ (xm , X
⊆
kϕfree(X) [x1 /ϕx1 , . . . , xm /ϕxm ]kT ϕ
By Lemma 1.19.4 and because in T ϕ we have that λ(xi ) = kϕxi kT ϕ , we obtain
kϕfree(X) [x1 /ϕx1 , . . . , xm /ϕxm ]kT ϕ ⊆ kϕfree(X) kT ϕ .
Since by Lemma 2.14.2 we have that kϕfree(X) kT ϕ = kϕkT we get this inclusion.
“⊆”: Let s ∈ kϕkT . By Theorem 1.28 there is a winning strategy in
E(ϕ, (T , s)) and by Theorem 2.25 it can be assumed to be normalized. Let
π be any play consistent with the strategy starting from hϕ, si. We have that
if there is a (first) regeneration of a ν-variable xi in a position hxi , si i then by
Lemma 2.28 we have that
si ∈ kNS+ (xi , X −i )kT
where X is the set of all ν-variables in ϕ. Therefore, there is a winning strategy
for Player 0 in
E(ϕfree(X) , (T [x1 7→ kNS+ (x1 , X −1 )kT , . . . , xn 7→ kNS+ (xn , X −n )kT ], s))
By Theorem 1.28 we have that
s ∈ kϕfree(X) kT [x1 7→kNS+ (x1 ,X −1 )kT ,...,xn 7→kNS+ (xn ,X −n )kT ]
and with Lemma 1.19.1 we complete the proof.
Corollary 2.33. The modal µ-calculus hierarchy on finite transitive systems
collapses to ∆µ2 .
tf
tf
tf
tf
Proof. By Theorem 2.32, ΣµT
= ∆µT
. By duality, ΠµT
= ∆µT
. By
2
2
2
2
this fact it is therefore very easy to verify inductively that for every n > 0,
tf
µTtf
µTtf
.
ΣµT
2+n = Π2+n = ∆2
Corollary 2.34. The modal µ-calculus hierarchy on transitive systems collapses to ∆µ2 .
Proof. Suppose that the hierarchy does not collapse. Therefore, there is a formula ϕ such that for all formula ψ ∈ ∆µ2 there is a transitive system T such that
T , s0 |= ¬(ϕ ↔ ψ). By Theorem 2.5, there is a finite transitive model T f such
that T f , sfi |= ¬(ϕ ↔ ψ). But this cannot be the case by Corollary 2.33.
We end with the definition of a syntactical translation from Lµ to ∆µ2 preserving equivalence on transitive transition systems.
Definition 2.35. R : Lµ → ∆µ2 is defined as
• R(p) = p and R(¬p) = ¬p
• R(⊥) = ⊥ and R(⊤) = ⊤
2.4. THE TRANSITIVE CASE
71
• R(α ◦ β) = R(α) ◦ R(β), where ◦ ∈ {∧, ∨}
• R(△ β) =△ R(β), where △∈ {, 3}
• R(µx.ϕ) = wn ρ(τ (wn(µx.(R(ϕ)))))
• R(νx.ϕ) = ¬(R(µx.¬ϕ[x/¬x]))
Lemma 2.36. For all µ-formula ϕ we have that
(1) R(ϕ) is well-defined, and
(2) R(ϕ) ∈ ∆µ2 .
Proof. We prove both parts simultaneously by induction on the structure of ϕ.
The induction cases for boolean and modal connectives are trivial.
If ϕ is of
the form µx.α we have that R(µx.α) = wn ρ(τ (wn(µx.(R(α))))) . Because τ
is a well-defined syntactical transformation, and neither wn nor τ increase the
alternation depth of a formula, the application of ρ in the clause of R(µx.α) is
well-defined by induction hypothesis. Thus, R(ϕ) terminates and therefore it is
well-defined too. The fact that R(µx.α) ∈ ∆µ2 follows by induction hypothesis,
by the fact that, by Remark 2.31, for all Σµ2 -formulae ψ we have that ρ(ψ) ∈ ∆µ2 ,
and because we know that τ and wn do not increase the alternation depth. If
ϕ is of the form νx.α, on one hand R(νx.α) is well-defined because the clause
for this form is defined via a reducing case R(µx.¬ϕ[x/¬x]), and, on the other
hand R(ϕ) ∈ ∆µ2 because ∆µ2 is closed under negation.
Theorem 2.37. For all ϕ ∈ Lµ and all finite transitive transition systems T we
have that
kϕkT = kR(ϕ)kT .
Proof. We prove the equivalence by induction on rank(ϕ) simultaneously for
all finite transitive transition systems T . The induction cases for boolean and
modal connectives are trivial. If ϕ is of the form µx.α we have that
kR(µx.α)kT = kwn ρ(τ (wn(µx.R(α)))) kT by definition of R
= kwn(µx.R(α))kT
τ (wn(µx.R(α))) ∈ Σµ2 , and by
Lemma 1.20 and Theorem 2.32
= kµx.αkT
by Lemma 1.20 and induction
hypothesis
If ϕ is of the form νx.α we do a similar induction step like above by using the
equivalence kνx.αkT = k¬µx.¬α[x/¬x]kT .
We conclude by verifying that the syntactical translation R is also an explicit
syntactical translation of all modal µ-formulae to the alternation free fragment
preserving denotation in every transitive transition systems. The proof goes
with similar argument as in Corollary 2.34 and it is left to the reader.
Theorem 2.38. For all ϕ ∈ Lµ and all transitive transition systems T we have
that
kϕkT = kR(ϕ)kT .
72
CHAPTER 2.
Remark 2.39. Note, that due to the example of Visser in [122] mentioned in the
introduction the alternation-free fragment is also the optimal bound if restrict
ourselves to transition systems which are transitive and reflexive.
Example 2.40. Let’s have a look at our example from Section 3.4. In the case
of “always eventually”, we have that
t
t
kνx.(µy.(p ∨ 3y)) ∧ x)kT = k(p ∨ 3p) ∧ (p ∨ 3p)kT .
For “infinitely often”, it holds that
t
t
kνx.µy.((p ∨ 3y)) ∧ 3x)kT = kνx.(p ∧ 3x)kT .
But, because from footnote 4 of the introduction we know that νx.(p ∧ 3x)
cannot be reduced to any purely modal formula, contrary to the transitive and
symmetric case, over transitive transition systems “infinitely often” cannot be
expressed by a ∆µ1 formula.
2.5
The reflexive case
In this section we prove the strictness of the modal µ-calculus hierarchy on
reflexive transition systems. This is done by following the argumentation of the
proof of the strictness of the hierarchy on all binary transition systems of Arnold
in [7]. First, we adapt the game transition system, introduced in Section 1.7,
such that it is reflexive.
Let E(ϕ, (T , s)) be a parity game with priority function Ω and with corresponding pointed game transition system T (E(ϕ, (T , s))). We extend the edge
relation E of the parity game to its reflexive closure E r = E∪{(s, s); s ∈ V0 ∪V1 },
and change our priority function Ω to Ωr such that for all vertices hψ, si where
ψ ≡ ηx.δ (η ∈ {µ, ν}) we have
Ωr (hψ, si) = Ω(hψ, si) + 2
and such that for all other vertices we define:
• if min Ω is even
Ωr (hψ, si) =
(
0 if hψ, si ∈ V1
1 if hψ, si ∈ V0 .
Ωr (hψ, si) =
(
2 if hψ, si ∈ V1
1 if hψ, si ∈ V0 .
• if min Ω is odd
The new resulting “reflexive” parity game is denoted as E r (ϕ, (T , s)). The
following Lemma can be proved by unwinding the definition of winning strategy.
Lemma 2.41. Player 0 has a winning strategy for E r (ϕ, (T , s)) iff Player 0 has
a winning strategy for E(ϕ, (T , s)).
Given a “reflexive” parity game E r (ϕ, (T , s)) the pointed game transition
system T (E r (ϕ, (T , s))) is defined analogously as above. Obviously, the pointed
game transition system T (E r (ϕ, (T , s))) is reflexive. We have that
73
2.5. THE REFLEXIVE CASE
Proposition 2.42. Let (T , s) be an arbitrary pointed transition system. For all
ϕ ∈ Πµn we have that
T (E r (ϕ, (T , s))) ∈ kWΠµn+2 k
if and only if
T (E(ϕ, (T , s))) ∈ kWΠµn k
and dually for ϕ ∈ Σµn .
Proof. This follows directly by the definition of the “reflexive” parity game
E r (ϕ, (T , s)) and by applying Proposition 1.32 to Lemma 2.41.
Corollary 2.43. Let (T , s) be an arbitrary pointed transition system. For all
ϕ ∈ Πµn we have that:
T (E r (ϕ, (T , s))) ∈ kWΠµn+2 k if and only if (T , s) ∈ kϕk.
and dually for ϕ ∈ Σµn .
Proof. By Proposition 2.42 and Corollary 1.34 we obtain our result.
For all formulae ϕ we define a function fϕ (functional class) mapping a
pointed transition system (T , s) to a reflexive transition system fϕ (T , s) such
that
fϕ (T , s) := T (E r (ϕ, (T , s))).
The proof of the next Lemma follows similar arguments as the proof of the
same result for the class of all binary trees proved by Arnold in [7], and which
has bee extended to arbitrary transition systems by Alberucci in [1].
Lemma 2.44. For all formulae ψ ∈ Σµn (resp. Πµn ), n ∈ N, there is an equivalent
formula ϕ ∈ Σµn (resp. Πµn ) such that the function fϕ has a fixpoint in Tr , that
is, a pointed reflexive transition system (T F , sF ) such that
fϕ (T F , sF ) = (T F , sF ).
Proof. First we remark that for any two pointed transition systems (T , s) and
(T ′ , s′ ) which are identical up to depth m we have that fψ (T , s) and fψ (T ′ , s′ )
are identical up to depth m. This can be proved by induction on the structure
of the formulae ψ and the proof is left to the reader. In order to prove the
Lemma let ψ be an arbitrary formula. We define ϕ ≡ ψ ∧ ψ. By the first remark
and the fact that fϕ (T , s) is basically the transition system fψ (T , s) with a new
distinguished state hψ ∧ ψ, si from where you can reach the distinguished state
of fψ (T , s) which is hψ, si. We get the following
Claim: Given two pointed transition systems (T , s) and (T ′ , s′ ) which are
identical up to depth m then fϕ (T , s) and fϕ (T ′ , s′ ) are identical up to depth
m + 1.
For the construction of the fixpoint (T F , sF ) we proceed as follows. We
start from a one state pointed transition system (T , s) = ({s}, {(s, s)}, λ) where
the valuation λ can be chosen arbitrarily. And define a sequence of pointed
transition systems {(T , s)n }n∈N as
(T , s)0 = (T , s) and (T , s)n+1 = fϕ ((T , s)n ).
By the claim above {(T , s)n }n∈N is a monotone sequence of reflexive transition
systems and it can easily be seen that beside being well defined lim({(T , s)n }n∈N )
is a reflexive transition system. It is an easy exercise to show that we have
fϕ (lim({(T , s)n }n∈N )) = lim({(T , s)n }n∈N ).
74
CHAPTER 2.
Theorem 2.45. For all natural numbers n ∈ N \ {0} we have that
r
r
ΣTn ( ΣTn+1
and
r
r
ΠTn ( ΠTn+1 .
Proof. We proof the contrapositive. Assume that we have
r
r
r
ΣTn+1 ⊆ ΣTn
r
or ΠTn+1 ⊆ ΠTn .
r
r
Without restriction of generality, assume ΣTn+1 ⊆ ΣTn . Then, if kϕk ∈ Πµn+1
we have k¬ϕk ∈ Σµn+1 and by assumption k¬ϕk ∈ Σµn and therefore kϕk ∈ Πµn .
Therefore, assuming the contrapositive leads to
r
r
r
ΣTn+1 ⊆ ΣTn
r
r
and ΠTn+1 ⊆ ΠTn .
r
r
r
Since from ΣTn+1 ⊆ ΣTn , by definition, it can be inferred that ΠTn ⊆ ΣTn , and
r
r
r
r
from ΠTn+1 ⊆ ΠTn , by definition, it can be inferred that ΣTn ⊆ ΠTn , by assuming
r
r
r
r
the contrapositive we get that ΠTn+1 = ΠTn = ΣTn+1 = ΣTn and, obviously, we
then have for all k ∈ N that
r
r
r
r
ΠTn+k = ΠTn = ΣTn+k = ΣTn .
(2.4)
Since WΣµn+2 ∈ Σµn+2 we have that ¬WΣµn+2 ∈ Πµn+2 and with equation 2.4 we
get
r
r
k¬WΣµn+2 kT ∈ ΣTn .
By Lemma 2.44 there is a formula ϕ ∈ Σµn equivalent to ¬WΣn+2 and a pointed
transition system (T F , sF ) such that
(T F , sF ) = fϕ (T F , sF ).
Since fϕ (T , s) is defined as T (E r (ϕ, (T , s))), by Corollary 2.43, for all pointed
transition systems (T , s) we have that fϕ (T , s) ∈ kWΣn+2 k if and only if (T , s) ∈
kϕk. Since ϕ is equivalent to ¬WΣn+2 we get that
(T F , sF ) ∈ k¬WΣn+2 k
iff (T F , sF ) ∈ kWΣn+2 k
which is a contradiction.
Theorem 2.46.
(1) The modal µ-calculus hierarchy is strict over reflexive transition systems.
(2) The modal µ-calculus hierarchy is strict over finite reflexive transition systems.
r
r
Proof. Part 1 is a corollary of Theorem 2.45. For Part 2, let kϕk ∈ ΣTn \ ΠTn .
Then, by Part 1 we know that for every ψ ∈ Σµn−1 it holds that ¬(ϕ ↔ ψ) has
a reflexive model. By Theorem 2.3, this model can be finite. Hence ϕ ∈ Σµn is
not equivalent to any Σµn−1 formula on finite reflexive transition systems.
75
2.6. SUMMARIZING REMARKS
2.6
Summarizing remarks
Thanks to the use of least and greatest fixpoint operators, the modal µ-calculus
is a powerful logic which can express many interesting properties of models.
With least fixpoints one can express, for instance, liveness properties like “it is
possible to reach a node where p holds”, and with greatest fixpoints one expresses
safety properties of the kind “p is true in all reachable nodes”. On arbitrary
transition systems, these two properties really need fixpoints in order to be
expressed. That is to say, they are not modally expressible. This is because
modal logic can only speak about local properties, in the sense that whether
or not a modal formula is true in a certain node of a model only depends on
the nodes accessible to the current one. But what gives to the µ-calculus all
its power is the possibility of having nested least and greatest fixpoints. For
instance, by one nesting it is possible to already capture fairness properties like
“there is a branch where p holds infinitely often”, while with several alternations
one can even express the existence of a winning strategy in a parity game.
Although on arbitrary transition systems the number of alternations between
different fixpoints generates a strict infinite hierarchy, the fixpoint hierarchy may
not be infinite anymore if we restrict the semantics to subclasses of models. In
this chapter, we were interested in the behavior of the modal µ-calculus on the
class of transitive models, of reflexive models, and on the class of symmetric
and transitive models. The obtained results are summarized in the next figure:
reflexive
fixpoint
alternation
hierarchy
strict
transitive
collapse to the
alternation
free fragment
symmetric & transitive
collapse to the
modal
fragment
The natural missing case is the case of symmetric models. We conjecture that
on this subclass of models the fixpoint alternation hierarchy is strict. Another
missing prominent subclass, which is probably the most studied so far, is the
class of transitive and upward well-founded models, also called the Gödel-Löb
class in view of its relation with Gödel theorems and the logic of provability in
Peano Arithmetic. This class is presented and studied in the next chapter.
Concerning the hierarchy on transitive models, d’Agostino and Lenzi [44]
propose a different proof of its collapse which explicitly uses Theorem 2.17 of
this thesis. This result has also independently been obtained by Dawar and
Otto [46].
76
CHAPTER 2.
Chapter 3
The µ-Calculus vs the
Gödel-Löb Logic
This chapter is based on a joint work with Luca Alberucci [4].
3.1
Preliminary remarks
The Gödel-Löb logic, GL for short, is a modal logic where the modal operator for
necessity is interpreted as provability in a reasonably rich formal theory such as
Peano arithmetic, and is thence used to investigate what arithmetical theories
can express in a restricted language about their provability predicates. This
modal logic has been studied since the early seventies, and has had important
applications in the foundations of mathematics ([30, 113]). As a formal system,
GL is obtained by adding the modal version of what is called the Löb’s theorem to
the minimal modal logic K. Beside the arithmetical interpretation there is also
a semantics given by transition systems. The class of all transitive and upward
well-founded systems, that is where there is no infinite chain of successor nodes,
forms a complete semantics for GL: a formula is derivable in the system GL if
and only if it is valid over the class of all transitive and upward well-founded
models.
Fixpoints and fixpoint theorems play an important role in GL. The most
famous one, the existence of a fixpoint for guarded formulae was proved by De
Jongh and Sambin independently (see [113]). Even though it is formulated and
proved by strictly modal methods, the fixpoint theorem still has great arithmetical significance. The uniqueness of the fixpoint was proved later by Bernardi,
De Jongh and Sambin independently (see [113]).
Since the modal µ-calculus is a general framework to study fixpoints in modal
logic, applying methods from the modal µ-calculus for the study of this modal
logic is a promising work. This has been done by van Benthem in [13] and
Visser in [123]. Both authors establish, by using the De Jongh-Sambin fixpoint
theorem, that the modal µ-calculus over GL collapses to its modal fragment.
But since they use the already known fixpoint theorem in order to establish this
collapse, in [13] van Benthem writes:
“Our [. . . ] analysis does not explain why provability fixed-points
77
78
CHAPTER 3.
are explicitly definable in the modal base language. Indeed, the
general reason seems unknown.”
In this chapter an answer to this question is given. More precisely, we prove the
collapse of the modal µ-calculus over GL without using the De Jongh-Sambin
Theorem by showing that fixpoints are reached after two iterations of wellnamed fixpoint formulae.
Fixpoint theorems in GL hold also for modal formulae where the variable
appears guarded but not necessarily positively and, from this point of view, this
first result is not completely satisfactory since modal µ-calculus allows fixpoint
constructors only for syntactically positive propositional variables. Therefore,
we also introduce the modal µ∼ -calculus which allows fixpoint constructors for
formulae where the fixpoint variable appears guarded. As can be done also
for the standard µ-calculus we define the semantics by way of games, in this
case only over transitive and upward well-founded transition systems. By using
game-theoretical arguments and providing an explicit syntactical translation of
the modal µ∼ -calculus into GL which preserves logical equivalence, we show that
the modal µ∼ -calculus collapses to the modal fragment. As a corollary of the
collapse, we obtain a new version of the De Jongh-Sambin Fixpoint Theorem
with a simple algorithm which shows how the fixpoint can be computed. In
this sense we give an answer to a generalization of van Benthem’s question.
Summing up, the modal µ∼ -calculus allows us to apply techniques similar as
those known from the standard µ-calculus to GL and could be regarded as a
starting point for further studies in this direction.
Both the collapse of the modal µ-calculus over GL and the one of the µ∼ calculus over the same class of models are proved by using techniques and results
from the previous chapter where the collapse of the semantical fixpoint alternation hierarchy over transitive models was proved.
In the next section we formally introduce Gödel-Löb Logic GL and some
results which are already known. In Section 2 we analyze the modal µ-calculus
over GL and show that it collapses to the modal fragment. In the last section we
introduce the modal µ∼ -calculus and show a collapse to the modal fragment.
The result is then used to provide a new proof of the uniqueness theorem of
Bernardi, De Jongh and Sambin and of the existence theorem of De Jongh,
Sambin. For the last one we also give a simple algorithm which shows how the
fixpoint can be computed.
3.2
Gödel-Löb Logic GL
3.2.1
Syntax and Semantics
We start from an infinite countable set Prop of propositional variables. Then the
collection LGL of GL-formulae is given by the usual grammar for modal formulae:
ϕ ::= p | ∼p | ⊤ | ⊥ | (ϕ ∧ ϕ) | (ϕ ∨ ϕ) | 3ϕ | ϕ
where p ∈ Prop. If all propositional variables occurring in ϕ are in P ⊆ Prop,
we write ϕ ∈ LGL (P). If ψ is a subformula of ϕ, we write ψ ≤ ϕ. We write
ψ < ϕ when ψ is a proper subformula. sub(ϕ) is the set of all subformulae of ϕ.
The formula ¬ϕ is defined by using de Morgan dualities for boolean connectives
79
3.2. GÖDEL-LÖB LOGIC GL
and the modal dualities for 3 and and the law of double negation. As
usual, we introduce implication ϕ → ψ as ¬ϕ ∨ ψ and equivalence ϕ ↔ ψ as
(ϕ → ψ) ∧ (ϕ → ψ). We say that p ∈ Prop is guarded in ϕ if p ≤ ϕ and all
occurrences of p are in the scope of a modal operator.
The axioms and inference rules below give a deduction system for GL. As
usual we write GL ⊢ ϕ if there is a derivation of ϕ in the system presented below.
Axioms: All classical propositional tautologies, the Distribution Axiom from
modal logic and the Löb Axiom
(α → α) → α.
Inference Rules are the Modus Ponens and the Necessitation Rule.
As for all modal logics the semantics of GL is given by transition systems.
Recall that a transition system T is of the form (S, →T , λT ) where S is a set
of states, →T is a binary relation on S called the accessibility relation and
λ : Prop → ℘(S) is a valuation for all propositional variables. A transition
system T with a distinguished state s is called a pointed transition system and
denoted by (T , s). T denotes the class of all pointed transition systems. The
accessibility relation is called upward well-founded if there is no infinite chain of
the form
s0 →T s1 →T s2 → . . . .
By Twft we denote the subclass of pointed transition systems such that the
accessibility relation is transitive and upward well-founded.
The denotation of a GL-formula in a transition system T and the notion of
being a model of a GL-formula are defined as for modal logic. When all pointed
transition systems (T , s) ∈ Twft are a model of ϕ, we write GL |= ϕ. A proof of
the next theorem can be found in [30].
Theorem 3.1. For all GL-formulae ϕ we have that
GL ⊢ ϕ
3.2.2
if and only if
GL |= ϕ.
Embedding GL into the modal µ-calculus
In this subsection we present an embedding t from GL into the modal µ-calculus.
First, we define the function ()∗ : LGL (P) → Lµ (P) recursively on the structure
of the formula such that
• (p)∗ ≡ p and (∼p)∗ ≡∼p,
• (α ∧ β)∗ ≡ (α)∗ ∧ (β)∗ and (α ∨ β)∗ ≡ (α)∗ ∨ (β)∗
• (α)∗ ≡ νx.(x ∧ (α)∗ ), and
• (3α)∗ ≡ µx.3(x ∨ (α)∗ ).
The embedding t : LGL (P) → Lµ (P) is now defined as
t(ϕ) ≡ (µx.x) → (ϕ)∗ .
The following theorem is due to van Benthem [13]. It shows that GL which
semantically lives on transitive and upward well-founded transition systems can
be translated into the modal µ-calculus over arbitrary transition systems. For
the first equivalence van Benthem provides a syntactical proof without using
completeness results.
80
CHAPTER 3.
Theorem 3.2 ([13]). For all formulae ϕ ∈ LGL we have that
(GL ⊢ ϕ
⇔
Koz ⊢ t(ϕ))
and
(GL |= ϕ
⇔
|= t(ϕ)).
The next lemma verifies that over upward well-founded transition systems,
least fixpoints and greatest fixpoints coincide.
Lemma 3.3. Let T be an upward well-founded transition system. Then, for
every ϕ(x) ∈ Lµ such that x is guarded and positive it holds that
kµx.ϕ(x)kT = kνx.ϕ(x)kT .
Proof. Note, that in an evaluation game there are no infinite regeneration of x
since then we would have an infinite chain of the form
s0 →T s1 →T s2 . . . .
Therefore, a winning play for νx.ϕ is also a winning play for µx.ϕ. With the
Fundamental Theorem of the semantic of the modal µ-calculus (Theorem 1.28)
we get the result.
3.3
The modal µ-calculus over GL
In this section we show that the expressivity of the modal µ-calculus over GL,
that is, over transitive and upward well-founded transition systems, is the same
as the one of the modal base language. In this sense we answer to van Benthem’s
question cited in the introduction.
In the previous chapter we showed that over transitive transition systems
every µ-formula is equivalent to a µ-formula without alternation of fixpoint operators. Moreover, we showed that under certain conditions a fixpoint operator
can be eliminated by regenerating the formula. More precisely, in Theorem
2.17 we proved that given a transitive transition system T , and a well-named
µ-formula ϕ(x) such that x ∈ free(ϕ) and occurs only once:
(1) If x is in the scope of a in νx.ϕ(x) then
kνx.ϕ(x)kT = kϕ(ϕ((⊤))kT .
(2) If x is in the scope of a 3 in µx.ϕ(x) then
kµx.ϕ(x)kT = kϕ(ϕ((⊥))kT .
Definition 3.4. The translation τ : Lwn
µ (P) → LGL (P) is defined recursively on
the rank of the formula such that τ ((∼)p) ≡ (∼)p, such that τ distributes over
boolean and modal connectives and such that for all η ∈ {µ, ν} we have
(
τ (wn(ϕ(ϕ(⊤)))) x is in the scope of a in ϕ,
τ (ηx.ϕ) =
τ (wn(ϕ(ϕ(⊥)))) else.
Obviously, by first well-naming a formula and then applying τ we get a translation from Lµ (P) to LGL (P).
3.4. THE MODAL µ∼ -CALCULUS
81
The fact that over well-founded transition systems greatest and least fixpoint
coincide almost immediately leads us to the collapse of the modal µ-calculus over
GL into its modal fragment.
Theorem 3.5. On transitive and upward well-founded transition systems we
have that the following holds for every ϕ ∈ Lµ :
kϕkT = kτ (wn(ϕ))kT .
Proof. By Lemma 1.18 we can assume that ϕ is well-named. The proof is by
induction on rank(ϕ). The base case and the case where rank(ϕ) is a successor
ordinal are straightforward. If rank(ϕ) is a limit ordinal then ϕ is of the form
ηx.α (η ∈ {µ, ν}). Assume that ϕ is of the form νx.ϕ. If x is in the scope of
a in ϕ then the induction step follows from Theorem 2.17. Else, x is only in
the scope of some 3 in ϕ. In this case by Lemma 3.3 we have that
kνx.ϕkT = kµx.ϕkT
and by applying once more Theorem 2.17 we get the induction step. The case
where ϕ is of the form µx.ϕ is shown by analogous arguments.
3.4
The modal µ∼ -calculus
In this section we introduce a new language, called the modal µ∼ -calculus,
which, in some sense, can be seen as an extension of the guarded fragment of
the modal µ-calculus. The main novelty is that we allow the µ-operator to bind
negative (and guarded) occurrences of propositional variables. Therefore, the
modal µ∼ -calculus allows us to refer explicitly, that is, in a µ-calculus style,
to fixpoints of guarded formulae. For example, the fixpoint of the “equation”
p ↔ α(p) where α(x) is a guarded formula can be directly denoted as µx.α(x).
As it can be done for the modal µ-calculus the semantics of the modal µ∼ calculus is defined by way of games over transitive and upward well-founded
transition systems. We provide an explicit syntactical translation of the modal
µ∼ -calculus into GL which preserves logical equivalence. As a corollary of the
collapse, we obtain a new version of the De Jongh-Sambin Fixpoint Theorem.
The modal µ∼ -calculus could be seen as a starting point for the application of
tools of the standard µ-calculus, as for example games, to GL.
3.4.1
Basic notions and results
The language of the modal µ∼ -calculus, Lµ∼ , is almost the same as the one for
the modal µ-calculus with the only difference that we allow fixpoint constructors
also when the bound variable is appearing negatively, that is, modal µ∼ -formulae
(or simply µ∼ -formulae) are defined as follows:
ϕ ::= p | ∼p | ⊤ | ⊥ | (ϕ ∧ ϕ) | (ϕ ∨ ϕ) | 3ϕ | ϕ | µx.ϕ
where p, x ∈ Prop and where x appears guarded in ϕ. All syntactical notions,
such as Lµ∼ (P), bound variable, rank of a formula, ϕx , ϕfree(X) etc., are defined
as for the modal µ-calculus. Without loss of generality, we always suppose that
bound(ϕ) ∩ free(ϕ) = ∅.
We say that a µ∼ -formula ϕ is in normal form if bound(ϕ) ∩ free(ϕ) = ∅ and
if for all subformulae of ϕ of the form µx.µy.α we have that
82
CHAPTER 3.
• α is not of the form µz.β, and
• x occurs only negatively in α and y has only positive occurrences in α.
For the substitution, if x ∈ free(ϕ), then ϕ[x/ψ] is given by substituting
¬ψ to every negative occurrence ∼ x and by substituting ψ to every positive
occurrence x. As for the modal µ-calculus negation is defined by using de
Morgan laws and the duality of and 3, in addition we set
¬µx.α ≡ µx.¬α[x/¬x].
The last equivalence can be rather surprising at a first look. It is motivated by
the fact that the modal µ∼ -calculus will be interpreted over upward well-founded
models, where least and greatest fixpoint coincide.
Note that, for every y ∈ bound(µx.α), y is negative in µx.α if and only if y
is negative in ¬µx.α.
The semantics for the modal µ∼ -calculus over GL is given by evaluation
games on pointed upward well-founded and transitive transition systems. These
evaluation games are similar to the ones for the modal µ-calculus. Let ϕ ∈ Lµ∼
and (T , s) ∈ Twft .
• First we construct recursively the two arenas hV0+ , V1+ , E + i from ϕ and
(T , s) and hV0− , V1− , E − i from ¬ϕ and (T , s) as it is done for the modal
µ-calculus, except the fact that, for each vertex of the form h∼x, ti which
was generated in the recursion defining the arena hV0+ , V1+ , E + i we put
the condition
h∼x, ti ∈ V0+ and E + (h∼x, ti) = ∅,
and if it was generated in the recursion defining the arena hV0− , V1− , E − i
we set
h∼x, ti ∈ V0− and E − (h∼x, ti) = ∅.
• Then the arena of E(ϕ, (T , s)) is the triple hV0 , V1 , Ei defined by taking the
disjoint union of the two previous arenas, with the following modification:
– For every vertex of the form h∼x, ti where x ∈ bound(ϕ) we set
(
{h¬ϕx , ti} ⊆ V − if h∼x, ti ∈ V0+
E(h∼ x, ti) =
{hϕx , ti} ⊆ V +
if h∼x, ti ∈ V0− .
Since we are on upward well-founded models and that all regenerated variables
are guarded, all plays are finite. Therefore, we have that Player 0 wins if and
only if the last vertex of the play belongs to Player 1. Because we do not have
to care about priorities, the previous winning condition is admissible and the
evaluation game for µ∼ -formulae is therefore well-defined1 .
We say that a pointed upward well-founded transitive transition system
(T , s) is a model of a µ∼ -formula if and only if Player 0 has a winning strategy
in E(ϕ, (T , s)). Further, we define
kϕkW
T = {s ∈ S | (T , s) is a model of ϕ}.
1 Note that on non well-founded models a play can be infinite. Thus, since in this kind of
plays it is possible that Player 0 and Player 1 “switch” their roles infinitely often, it is not
clear how to extend our game-theoretical approach also to non well-founded models by adding
a natural and uniform (parity) winning conditions for infinite plays.
3.4. THE MODAL µ∼ -CALCULUS
83
By kϕkW we denote the class of all upward well-founded and transitive models
of ϕ, that is, all pointed transition systems (T , s), transitive and upward wellfounded, such that s ∈ kϕkW .
Example 3.6. Consider the formula µx.3 ∼x. This formula says that Player 0
can always force the number of the states visited in a play to be even. Because
the considered models are transitive, this implies that the formula says that the
root of the models has at least one accessible state.
The next lemma states some basic properties of denotation.
Lemma 3.7. For all transition systems T = (S, →T , λT ) and all µ∼ -formulae
µx.ϕ we have that
W
(1) kµx.µy.ϕ(x, y)kW
T = kµx.ϕ(x, x)kT ,
W
(2) kµx1 . . . µxn .ϕkW
T = kµxp(1) . . . µxp(n) .ϕkT , where p is any permutation
over {1, . . . , n},
W
(3) There is a well-named formula wn(ϕ) such that kϕkW
T = kwn(ϕ)kT ,
(4) There is a formula nf(ϕ) in normal form such that kϕkT = knf(ϕ)kT .
Proof. Part 1 is by definition of the evaluation game for the modal µ∼ -calculus.
Part 2 is proved by an easy induction on the length of the prefix. Part 3 is a
straightforward consequence of part 1. Part 4 is a straightforward consequence
of part 1 and part 2.
The next lemma shows that over upward well-founded models, the positively
bounded fragment of the modal µ∼ -calculus coincides with the guarded fragment
of the standard modal µ-calculus.
Lemma 3.8. Let ϕ ∈ Lµ ∩ Lµ∼ . Then for every upward-well founded model T
kϕkW
T = kϕkT .
Proof. This follows by applying Theorem 1.28 to the fact that, for every ϕ ∈
Lµ ∩ Lµ∼ , the evaluation games for Lµ∼ and the evaluation games for the µcalculus coincide over upward well-founded models.
The next lemma shows that negation behaves as expected.
Lemma 3.9. Let ϕ be a µ∼ -formula and T = (S, →T , λT ) an upward wellfounded transition system. We have that
W
k¬ϕkW
T = S \ kϕkT .
Proof. Consider the evaluation game E(ϕ, (T , s)) where Player 0 starts to play
as Player 1 and vice versa. Clearly Player 0 (resp. Player 1) has a winning
strategy in this modified game iff she has a winning strategy in E(¬ϕ, (T , s)).
From this fact we get the claim.
The next lemma shows that in the modal µ∼ -calculus formulae of the form
µx.ϕ indeed define a fixpoint.
84
CHAPTER 3.
Lemma 3.10. For every µx.ϕ ∈ Lµ∼ and every upward well-founded transition
system T it holds that
W
kµx.ϕkW
T = kϕ[x/µx.ϕ]kT .
Proof. This result follows straightforwardly by definition of the evaluation game
for the modal µ∼ -calculus.
3.4.2
The unicity of fixpoints
Let T be a upward well-founded and transitive transition system and ϕ a µ∼ formula. Consider an arbitrary (memoryless) strategy σ for Player 0, not necessarily winning. We define the restriction of E(ϕ, (T , s0 )) on σ, denoted by
E|σ (ϕ, (T , s0 )), as follows:
• The set of positions V |σ of the restriction is given by all nodes which
are the positions of some play compatible with σ starting from position
hϕ, s0 i,
• The arena of E|σ (ϕ, (T , s0 )) is the triple hV0 |σ , V1 |σ , E|σ i where:
(1)
(2)
(3)
(4)
V0 |σ = ∅,
V1 |σ = V |σ ,
if hψ, si ∈ V |σ ∩ V1 then E|σ (hψ, si) = E(hψ, si), and
if hψ, si ∈ V |σ ∩ V0 then E|σ (hψ, si) = {σ(hψ, si)}.
We have that in E|σ (ϕ, (T , s0 )) the only Player who can move is Player 1.
This can be done because the moves for Player 0 are already completely determined by the (memoryless) strategy σ. Clearly, any play in E|σ (ϕ, (T , s0 )) is a
play in E(ϕ, (T , s0 )) compatible with σ. We say that a play π in E|σ (ϕ, (T , s0 ))
is winning for Player 0 if and only if the play π is winning for Player 0 in
E(ϕ, (T , s0 )). If σ is a winning strategy for Player 0 then any play in E|σ (ϕ, (T , s0 ))
is winning for Player 0.
In the next definition, we define a measure on the graph of an evaluation
game restricted to a strategy for Player 0, which is essentially just the distance
in the considered graph.
Definition 3.11. Let T be a upward well-founded transitive transition system,
ϕ a µ∼ -formula and σ any strategy for Player 0 in the parity game E(ϕ, (T , s0 )).
Then, for every position hψ, si of E|σ (ϕ, (T , s0 )), we define a measure d(hψ, si):
(
0
if E|σ (hψ, si) = ∅
d(hψ, si) =
′
′
sup{d(hψ, s i) + 1 : hψ, s i ∈ E|σ (hψ, si)} otherwise.
Note that, since T is upward well-founded, there cannot be an infinite chain
of the form ha0 , a1 , a2 , . . . i such that for every i ≥ 0, hai , ai+1 i ∈ E|σ . Therefore
for all evaluation games E(ϕ, (T , s0 )) and all vertices hψ, vi in the arena d(hψ, vi)
is a well-defined ordinal number, such that if a vertex hα, v ′ i is reachable from
a vertex hβ, v ′′ i then we have that d(hα, v ′ i) < d(hβ, v ′′ i).
The next theorem shows that a fixpoint formula µx.α(x) in the modal µ∼ calculus defines a fixpoint, as proved in Lemma 3.10, and that any other fixpoint
of a formula α(x) is identical to µx.α(x). In this sense it is an existence and
uniqueness theorem, and it is the central result of the section.
3.4. THE MODAL µ∼ -CALCULUS
85
Theorem 3.12. Let ϕ(x) ∈ Lµ∼ and x ∈ free(ϕ) a guarded variable. Let T be
a upward well-founded transitive transition system. Then, for every A ⊆ S we
have
W
kϕ(A)kW
T = A if and only if A = kµx.ϕ(x)kT .
Proof. The implication from right to left follows from Lemma 3.10. In order
to prove the implication from left to right, suppose kϕ(A)kW
T = A. From the
definition of evaluation game we straightforwardly can derive the inclusion A ⊆
W
kµx.ϕ(x)kW
T . For the other inclusion, suppose that s ∈ kµx.ϕ(x)kT . We have
that Player 0 has a winning strategy σ in E(µx.ϕ, (T , s)). Consider the restricted
evaluation game E|σ (µx.ϕ, (T , s)). For each vertex hα, vi in E|σ (µx.ϕ, (T , s)),
we have that d(hα, vi) is a well-defined measure. Clearly, with the following
claim we finish the proof.
Claim: For all vertices of the form hα, s′ i in E|σ (µx.ϕ, (T , s)) if α = µx.ϕ
then s′ ∈ A, and, if α = µx.¬ϕ then s′ ∈
/ A.
The proof of the claim is by induction on d. Since d(hµx.ϕ, s′ i) > 0 and
d(hµx.¬ϕ, s′ i) > 0 the induction base is trivial. For the induction step assume
first that we have a vertex hµx.ϕ, s′ i in E|σ (µx.ϕ, (T , s)). We distinguish two
cases:
(1) If from hµx.ϕ, s′ i there is no a reachable vertex of the form hµx.ϕ, s′′ i or
′
hµx.¬ϕ, s′′ i then we have that s′ ∈ kϕ(A′ )kW
T for all sets of states A and,
W
′
W
therefore, we also have s ∈ kϕ(A)kT . Since by assumption kϕ(A)kT = A
we proved the claim.
(2) Otherwise we distinguish two subcases given by the first vertex reached
which is of either the form hµx.ϕ, s′′ i or hµx.¬ϕ, s′′ i.
(a) If the first vertex reached of such kind is hµx.ϕ, s′′ i, then, since
have that d(hµx.ϕ, s′ i) > d(hµx.ϕ, s′′ i), by induction hypothesis
get s′′ ∈ A.
(b) If the first vertex reached of such kind is hµx.¬ϕ, s′′ i, then, since
have that d(hµx.ϕ, s′ i) > d(hµx.¬ϕ, s′′ i), by induction hypothesis
get s′′ ∈
/ A.
we
we
we
we
Therefore, for each play consistent with σ starting from hµx.ϕ, s′ i it holds
that if it reaches first a vertex of the form hµx.ϕ, s′′ i (or equivalently of
the form hx, s′′ i) we have that s′′ ∈ A, and, if it reaches first a vertex of
the form hµx.¬ϕ, s′′ i (or equivalently of the form h∼x, s′′ i) we have that
s′′ 6∈ A. But it can easily be seen that this implies s′ ∈ kϕ(A)kW
T . Since
by assumption kϕ(A)kW
T ⊆ A we finish the induction step for the case
α = µx.ϕ(x).
The induction step for a vertex of the form hµx.¬ϕ, s′ i is verified in the same way
W
by using the fact that by Lemma 3.9 we have that k¬ϕ(A)kW
T = S \ kϕ(A)kT
W
and, therefore, by assumption that k¬ϕ(A)kT = S \ A.
Corollary 3.13. Let ϕ and ψ be two µ∼ -formulae. If for all upward wellW
founded transitive transition system T we have that kψkW
T = kϕkT then for all
variables x and all T we have that
W
kµx.ψkW
T = kµx.ϕkT
86
CHAPTER 3.
Proof. By the “if” direction of Theorem 3.12 we have that
W
kµx.ψkW
T = kψkT [x7→kµx.ψkW ]
T
and with the premise of the corollary we get
W
kµx.ψkW
T = kϕkT [x7→kµx.ψkW ] .
T
Applying the “only if” direction of Theorem 3.12 we obtain that
W
kµx.ψkW
T = kµx.ϕkT .
The next theorem provides a new proof of Bernardi, De Jongh, Sambin
Theorem (c.f. Chapter 8 in [30] or [113]) using our results on the modal µ∼ calculus.
Theorem 3.14. Let ϕ(x) ∈ LGL , where x is guarded. We have that
GL ⊢ s (p ↔ ϕ(p)) ∧ s (q ↔ ϕ(q)) → (p ↔ q)
where s ϕ :≡ ϕ ∧ ϕ.
Proof. By Theorem 3.2 it is enough to show that for all ϕ(x) ∈ LGL it holds
that
|= µx.x → (((s )(p ↔ ϕ(p)) ∧ (s )(q ↔ ϕ(q))) → (p ↔ q))∗ .
And this can be done by showing for all ϕ(x) ∈ LGL that the following formula
is valid for all transition systems
µx.x → (p ↔ ϕ(p) ∧ q ↔ ϕ(q) ∧ ∗ (q ↔ ϕ(q)) ∧ ∗ (q ↔ ϕ(q)) → (p ↔ q))
(3.1)
where ∗ γ ≡ νx.(x ∧ γ).
Assume, that we have
s ∈ kµx.x ∧ p ↔ ϕ(p) ∧ q ↔ ϕ(q) ∧ ∗ (q ↔ ϕ(q)) ∧ ∗ (q ↔ ϕ(q))kT .
Then, (T , s) is well-founded and we have for s and for all reachable states s′
from s that q ↔ ϕ(q) and p ↔ ϕ(p). Therefore, if we assume that T consists
of s and all reachable states from s, which is an admissible assumption, we get
that we have
λT (p) = kϕ(λT (p))kW
T
and λT (q) = kϕ(λT (q))kW
T .
By Theorem 3.12 we get that
λT (p) = kµx.ϕ(x)kW
T
and λT (q) = kµx.ϕ(x)kW
T .
and therefore we obtain that
s ∈ kp ↔ qkT .
We have shown Equation 3.1 and finished the proof.
3.4. THE MODAL µ∼ -CALCULUS
3.4.3
87
Collapsing the modal µ∼ -calculus
In this subsection we provide an explicit syntactical translation of the modal µcalculus into GL which preserves logical equivalence. As a corollary, we obtain a
new proof of the De Jongh-Sambin Fixpoint Theorem which provides an explicit
construction of the fixpoint formula based on the syntactical translation defining
the collapse.
First of all, remember that, by Lemma 3.7.4, we can suppose that every
µ∼ -formula is in normal form.
Lemma 3.15. Let α(x) be a modal formula such that x appears only negatively
and guarded. Then, for every T ∈ Twft we have that
W
kµx.(α[x/α(x)])kW
T = kµx.α(x)kT .
Proof. Let A be kµx.α(x)kW
T . By the “if” direction of Theorem 3.12 we have
that kα(A)kW
=
A.
We
can
iterate this equivalence twice and get
T
kα[x/α(A)]kW
T = A.
Applying the “only if” direction of Theorem 3.12 gives us
kµx.((α[x/α(x)])kW
T =A
and therefore the proof of this lemma.
Note that, if x ∈ bound(µx.α) appears only negatively, then x occurs only
positively in µx.(α[x/α(x)]).
Everything is now set up in order to prove that the modal µ∼ -calculus over
GL collapses to its modal fragment.
Definition 3.16. The syntactical translation I : Lµ∼ → LGL uses the translation τ from Lµ to LGL of Definition 3.4. It is defined recursively as follows:
• I(p) = p and I(∼p) =∼p.
• I(⊥) = ⊥ and I(⊤) = ⊤.
• I(α ◦ β) = I(α) ◦ I(β), where ◦ ∈ {∧, ∨}.
• I(△ β) =△ I(β), where △∈ {, 3}.
• Assume that nf(µx.I(α(x))) is of the form µz.µy.α̂(z, y). We set
I(µx.α) = τ (wn(µz.β(z))),
where β(z) ≡ τ (wn(µy.α̂(z, y)))[z/τ (wn(µy.α̂(z, y)))]).
Lemma 3.17. The translation I is well-defined and, moreover, if
ϕ ∈ Lµ∼ (P) then I(ϕ) ∈ LGL (P).
88
CHAPTER 3.
Proof. By induction on the structure of the formula. The only critical case
is when ϕ ≡ µx.α. By induction hypothesis, I(α(x)) ∈ LGL . Therefore
α̂(y, z) ∈ LGL . By definition of normal form, z occurs only negatively and y occurs only positively in µz.µy.α̂(y, z). Thus, µy.α̂(y, z) ∈ Lµ . This implies that
wn(µy.α̂(y, z)) is well-defined and by Theorem 3.5 that τ (wn(µy.α̂(y, z))) ∈ LGL .
Note that y occurs only positively in α̂(y, z) and wn(µy.α̂(y, z)) is given by duplicating and renaming y. Therefore, it follows that z occurs only positively
in
τ (wn(µy.α̂(z, y)))[z/τ (wn(µy.α̂(z, y)))]).
This implies that µz.β(z) ∈ Lµ and therefore that wn(µz.β(z)) is well-defined.
Thence by Theorem 3.5 we have that τ (wn(µz.β(z))) ∈ LGL .
Theorem 3.18. Let ϕ ∈ Lµ∼ . On upward well-founded and transitive transition
systems T we have that
W
kϕkW
T = kI(ϕ)kT .
Proof. The proof goes by induction on rank(ϕ). If rank(ϕ) = 1 or rank(ϕ) is
a successor ordinal the induction step is straightforward. If rank(ϕ) is a limit
ordinal then ϕ is of the form µx.α. In this case by Lemma 3.7 we have that
W
kµx.I(α)kW
T = kµz.µy.α̂(z, y)kT .
W
Since by induction hypothesis we have that kI(α)kW
T = kαkT , with Corollary
3.13 we get that
W
kµx.I(α)kW
T = kµx.αkT
and therefore that
W
kµx.αkW
T = kµz.µy.α̂(z, y)kT .
(3.2)
Since by Lemma 3.17 and by construction of normal forms, α̂ is a modal formula
we have that µy.α̂ ∈ Lµ . With Theorem 3.5 and Lemma 3.7 we get that for all
upward well-founded and transitive T we have that
W
kµy.α̂kW
T = kτ (wn(µy.α̂))kT .
By Corollary 3.13 it holds that
W
kµz.µy.α̂kW
T = kµz.τ (wn(µy.α̂))kT
and with Equation 3.2 that
W
kµx.αkW
T = kµz.τ (wn(µy.α̂))kT .
(3.3)
Remember that y occurs only positively and z only negatively in α̂. Moreover
wn(µy.α̂(y, z)) is obtained by multiplying and renaming y. Therefore, since z appears only negatively in µy.α(y, z) it appears only negatively in wn(µy.α̂(y, z)),
too. Now, note that by definition of τ we are “regenerating” the formula only
on positive occurrences and, therefore, we have that z appears only negatively
in τ (µy.α̂), too. By Lemma 3.15 it holds that
W
kµz.(τ (wn(µy.α̂)))kW
T = kµz. τ (wn(µy.α̂))[z/τ (wn(µy.α̂))](z) kT .
With Equation 3.3 we get
W
kµx.αkW
T = kµz. τ (wn(µy.α̂))[z/τ (wn(µy.α̂))](z) kT .
By Lemma 3.7 and Theorem 3.5 we finish the induction step.
3.4. THE MODAL µ∼ -CALCULUS
89
The last theorem of this chapter is a new version of the De Jongh-Sambin
Fixpoint Theorem. Our version provides an explicit construction of the fixpoint
formula based on the definition of I.
Theorem 3.19. Let ϕ(x) ∈ LGL (P), where x is guarded. We have that
GL ⊢ I(µx.ϕ) ↔ ϕ(I(µx.ϕ)).
Further if ϕ ∈ LGL (P) then we have that I(µx.ϕ) ∈ LGL (P \ {x}).
Proof. The fact that I(µx.ϕ) ∈ LGL (P \ {x}) follows from Lemma 3.17. For the
provable equivalence, we show that GL |= I(µx.ϕ) ↔ ϕ(I(µx.ϕ)). The proof
then follows by Theorem 3.1. Let T ∈ Twft . We have
kI(µx.ϕ(x))kT =
=
=
=
=
=
kµx.ϕ(x)kW
T
kϕ(µx.ϕ(x))kW
T
kϕ(x)kW
T [x7→kµx.ϕkW
T ]
kϕ(x)kW
T [x7→kI(µx.ϕ)kW
T ]
kϕ(x)kT [x7→kI(µx.ϕ)kT ]
kϕ(I(µx.ϕ))kT
Lemma 3.8 and Theorem 3.18
Lemma 3.10
Definiton of evaluation game
Theorem 3.18
Lemma 3.8
Definition of denotation
We end with two examples where we apply our translation in order to solve
a modal equation.
Example 3.20. Consider the modal equation x ↔ ¬x. It is the same as
x ↔ 3 ∼x.
(3.4)
By Theorem 3.19 the µ∼ -formula µx.3 ∼x is the solution of Equation 3.4. By
definition of I we have that
I(µx.3 ∼x) = τ (µx.3¬3 ∼x) = τ (µx.3x) = 33⊤.
Note, that on upward-well-founded transitive transition system T , it holds that
k33⊤kT = k¬⊥kT .
Example 3.21. Consider the modal equation x ↔ ((x → q) → ∼x). This
is the same as
x ↔ 3(x∧ ∼q) ∨ ∼x.
(3.5)
By Theorem 3.19 the formula I(µx.3(x∧ ∼q) ∨ ∼x) is a solution of Equation
3.5. Let’s trace the construction of the fixpoint given by Definition 3.16:
We have that
α̂ ≡ 3(x∧ ∼q) ∨ ∼y
and that
τ (µx.α̂) ≡ 3((3(⊥∧ ∼q) ∨ ∼y)∧ ∼q) ∨ ∼y.
The formula τ (µx.α̂) can be simplified by using the following equivalence
kτ (µx.α̂)kT = k3( ∼y∧ ∼q) ∨ ∼ykT .
90
CHAPTER 3.
Now, we calculate β(y) of Definition 3.16 by using the simplified τ (µx.α̂) above
and get
β(y) ≡ 3(¬(3( ∼y∧ ∼q) ∨ ∼y)∧ ∼q) ∨ ¬(3( ∼y∧ ∼q) ∨ ∼y).
By definition of negation, we get
β(y) ≡ 3(((3y ∨ q) ∧ 3y)∧ ∼q) ∨ ((3y ∨ q) ∧ 3y).
Note that the following semantical equivalences hold
• k3(((3y ∨ q) ∧ 3y)∧ ∼q)kT = k3(⊥∧ ∼q)kT , and
• k((3y ∨ q) ∧ 3y)kT = k⊥kT .
Therefore, we get
kµy.β(y)kT = k3(⊥∧ ∼q) ∨ ⊥kT = k(⊥ → q) → ⊥kT .
Since I(µx.3(x∧ ∼ q) ∨ ∼ x) ≡ τ (wn(µy.β(y))) it follows that the formula
(⊥ → q) → ⊥ is a solution of Equation 3.5.
3.5
Summarizing remarks
An important theorem in the study of the expressive power of the modal µcalculus on restricted classes of transition systems is the De Jongh-Sambin Theorem. This theorem considers the modal logic GL, from Gödel-Löb, which besides an arithmetical interpretation, have also a complete semantics given by
the class of all transitive and upward well-founded systems. The theorem says
that fixpoint modal equations in GL have a unique solution. From this result it
follows that over the previous class of models, the µ-calculus collapses to modal
logic.
In this chapter, we decided to reverse this point of view. More precisely, we
proved the collapse of the modal µ-calculus over GL without using the De JonghSambin Theorem by showing that fixpoints are reached after two iterations of
well-named fixpoint formulae.
However fixpoint theorems in GL hold also for modal formulae where the
variable appears guarded but not necessarily positively. Thus, from this point
of view, the previous collapse is not completely satisfactory since modal µcalculus allows fixpoint constructors only for syntactically positive formulae. We
therefore have extended the collapse to an extension of the µ-calculus, called
the modal µ∼ -calculus, where fixpoint variables are not necessarily in positive
positions in the formulae. This was done by providing an explicit syntactical
translation of the µ∼ -calculus into GL which preserves logical equivalence. As a
corollary of this result, we have then obtained a new version of the De JonghSambin Fixpoint Theorem with a simple algorithm which shows how the fixpoint
can be computed.
Notice that Lemma 3.3 immediately leads to the fact that on finite trees the
modal µ-calculus collapses to the first ambiguous class of the fixpoint hierarchy.
Chapter 4
Characterizing the Modal
Fragment on Transitive
Models
This chapter is based on a joint work with Balder ten Cate [40].
4.1
Preliminary remarks
Determining effective characterizations of logics on trees is an interesting problem in theoretical computer science, the main motivation being the desire to
understand the expressive power of logics like first-order logic (FO), or the temporal logic CTL∗ , on trees.
For finite words, this problem is well-studied and understood. The first-order
definable regular languages, for instance, can be characterized as the class of
star-free regular languages [84] or the class of regular languages whose syntactic
monoid is aperiodic [109], a condition which can be tested effectively. Since
the regular languages are precisely the string languages definable in monadic
second-order logic with the child relation [36], this can also be seen as an effective characterization of FO as a fragment of MSO on strings. Similar effective
characterizations have been obtained for various fragments of FO and for various
temporal logics such as fragments of LTL (see [131] and [105] for references).
The list of results in the case of trees is much more frugal. The situation
seems to have improved a little in the last years, thanks to an effort towards
understanding fragments of CTL∗ and the successful use of what are called
forest algebra (see [29]). Notably, this formalism has been used for obtaining
decidable characterizations for the classes of tree languages definable in EF + EX
[28], EF + F−1 [20, 106], BC − Σ1 (<) [26, 106], ∆2 (≤) [27, 106]. This approach
has then been extended in the case of the temporal logic EF on infinite but
finitely branching trees by Bojanczyk and Idziaszek [24].
These results all demonstrate the importance of the algebraic approach in
obtaining decidable characterizations of logics. In the case of infinite trees, it is
natural to ask whether such logics also admit topological characterizations. Take
for instance the logic EF. It follows from the results of Bojanczyk and Idziaszek’s
91
92
CHAPTER 4.
[24] that the class of finite trees, that is the class of finitely branching tree models
of the formula µx.x, is not EF-definable (within the class of arbitrary finitely
branching trees). This is because the syntactic ω-forest algebra of this tree
language does not satisfy a certain equation. If we consider arbitrarily branching
trees, there is another explanation for the fact that the formula µx.x defines
a non-EF-definable tree language, which involves topology: µx.x defines the
class of well-founded trees, while EF formulae can only define tree languages
that are Borel. This raises the question whether EF, as a fragment of MSO, can
be characterized by topological means.
We give a positive answer in the case of EF. Specifically, we prove that a (not
necessarily finitely branching) tree language is EF-definable if and only if it is
invariant for EF-bisimilarity and Borel. Since EF is a fragment of weak monadicsecond order logic (WMSO) and all WMSO-definable tree languages are Borel,
we obtain as an immediate corollary that EF is the EF-bisimulation invariant
fragment of WMSO. Moreover, these characterizations are effective: given an
MSO formula, one can effectively test whether it defines an EF-definable tree
language. The proofs make crucial use of the results in [24]. As a corollary of
these characterizations of EF on arbitrary trees, we obtain that on transitive
transition systems, modal logic is the Borel fragment of the modal µ-calculus.
To sum up formally, we prove the following theorem:
Theorem 4.1. Let L be any MSO-definable tree language. The following conditions are equivalent and decidable:
(1) L is EF-definable
(2) L is WMSO-definable and closed under EF-bisimulation
(3) L is Borel and closed under EF-bisimulation
(4) L is closed under EF-bisimulation, and for every L-idempotent context c,
and for every forest f , c(f ) and (c + cf )∞ are L-equivalent.
The fourth condition, which is essentially the condition that was used in [24]
to characterize EF on finitely branching trees, involves some algebraic notions
that we will introduce in Section 4.2. Note that the previous equivalences do
not hold for finitely branching trees. This is because on finitely branching trees,
well-foundedness is equivalent to finiteness, which is Borel and closed under
EF-bisimulation but not EF-definable.
Finally, remark that since any EF-definable tree language is also definable in
first-order logic with the descendant relation, we obtain that all conditions of
Theorem 4.1 are equivalent to the condition that L is first-order definable and
closed under EF-bisimulation.
In the first part of Section 4.2, we introduce the basic notions of forest and
context, as well as an analogous to the well known Myhill-Nerode equivalence
relation but for tree languages. Then in its second part, after discussing some
properties of tree languages definable in monadic second order logic and defining
the logic EF, we present a characterization of this last formalism on finitely
branching trees by Bojanczyk and Idziaszek [24], which, as we point out, in fact
generalizes to arbitrarily branching trees. In Section 4.3 we present a natural
(prefix) topology for trees, and give some examples of Borel and non Borel
definable tree languages. The proof of Theorem 4.1 constitutes Section 4.4. In
4.2. PRELIMINARIES
93
the last section of this chapter we obtain as a corollary of the main Theorem
4.1 that on transitive transition systems, modal logic is the Borel fragment of
the modal µ-calculus.
4.2
4.2.1
Preliminaries
The beauty of forests
Forests, contexts. A forest over a finite set Σ is a sequence of conciliatory
trees over Σ. Formally, we represent a forest by a partial function f : N+ → Σ,
where N+ is the set of all non-empty sequences of natural numbers, such that
the domain of f is closed under non-empty prefixes. Given a forest f , and
x ∈ dom(f ), by f.x we denote the subtree of f rooted in x. If x ∈ N ∩ dom(f ),
the subtree f.x is called a rooted subtree of the forest f . With a slight abuse of
notation, we will sometimes identify trees with forests that have a single root.
A context is a forest with a hole, which is a leaf but not a root. Formally, a
context over Σ is a forest over the alphabet Σ ∪ {2} where exactly one node is
labeled by 2, and it is a leaf but not a root.
Operations on forests and contexts. We define two types of operations on
forests and contexts: a (horizontal) concatenation operation, denoted by “+”,
and a (vertical) composition operation, denoted by “·”. In spite of the notation
we use, these operations are in general not commutative.
Given a sequence of forests (fi : i ∈ α), with α ∈ ω ∪ {ω}, we want to
concatenate these forests. Note that each forest can contain infinitely many
rooted subtrees, and the length of the sequence itself, i.e., α, can be infinite.
However, for every i ∈ α, the set of all rooted subtrees of fi is countable, and
hence the set of all rooted subtrees of forests in the sequence is also countable.
Let (s′k : k ∈ ω) be any enumeration of the set of all rooted subtree of forests in
the sequence. If each forest fi consists of finitely many trees, we may assume that
the enumeration is in fact the natural enumeration induced by the order of the
sequence and the order on the rooted trees ofP
each forest. The concatenation
ofPthe forests fi is now defined as the forest i∈α fi where for every k ∈ ω,
( i∈α fi ).k = s′k .
We allow also to concatenate a sequence of forests with a context. Clearly
the result of such a concatenation is a context.
Concerning vertical composition, we want to compose a context p with a
forest, resp. a context, f , and obtain as a result a forest, resp. a context , c(t)
by replacing the hole node of c by f in some way. Since either the set of rooted
subtrees of f can be infinite, we distinguish between two cases. If this set is
finite, the composition c(f ) is just obtained by replacing the hole of c with the
forest f . Otherwise we proceed as follows. Let (si : i ∈ α) be any enumeration
of all the subtrees starting in a sibling of the hole node y of c, and let x be the
uniqueP
parent of y. Then the composition c(f ) is given by substituting in c the
forest i∈α si + f to the forest given by all the subtrees having as a root a son
of x.
Note that restricted to finitely branching trees, context and forests, the
operations of concatenation and composition correspond to the ones in [24].
94
CHAPTER 4.
Remark 4.2. Admittedly, the definition of concatenation of trees and forests,
and therefore also of vertical composition, is not nice. This is because it is
neither associative nor commutative. However, throughout this chapter, we essentially work with languages definable in monadic second order logic with the
child relation but not with an order on siblings. This implies that, when needed,
the reader can safely think of trees and forests as unordered trees and forests,
and thus of the operation of concatenation as an analogous of the set-theoretic
operation of union for multisets, which is both associative and commutative.
More formally, in the sequel we work in fact with an algebra consisting of equivalence classes under reordering of siblings (i.e., where two trees are equivalent
if they are isomorphic as relational structures in a signature without the order
relation on siblings) and the introduced operations are indeed well defined and
well behaved operations on equivalence classes.
Myhill-Nerode equivalence. Given a tree language L, we want to define a
notion of equivalence with respect to L, analogous to the well known MyhillNerode equivalence relation for finite words. Intuitively, two forests, or two
contexts, are L-equivalent if they “behave the same” with respect to L. The
precise definition of this equivalence relation is a delicate matter. It seems
tempting to define two forests f, f ′ to be equivalent if for every context c, c(f ) ∈
L if and only if c(f ′ ) in L, and similarly for contexts. However, it turns out
that, in the setting of infinite trees, it is more convenient to work with a slightly
finer grained notion of equivalence. The definition we use, which we will now
present, is essentially the one in [24] (we will comment on the precise relationship
between the two later).
The crucial notion here is that of a template. There are two kinds of templates, forest-templates and context-templates. A forest-template for an alphabet Σ is a forest over the alphabet Σ ∪ {⋆} in which one or more leafs (possibly
infinitely many) are labeled ⋆, and no non-leaf node is labeled ⋆. Similarly, a
context-template for Σ is a forest over the alphabet Σ ∪ {⋆} in which one or
more nodes (possibly infinitely many) are labeled ⋆. Intuitively, the occurrences
of ⋆ in a forest-template are placeholders for forests, and the occurrences of ⋆
in a context-template are placeholders for contexts.
In what follows, we will make use of the operation of replacing a subtree with
another forest. We will not give a formal, precise definition of this operation.
Just note that it can be defined in a straightforward manner by using horizontal
concatenation.
Thus, given a forest-template f over Σ∪{⋆} and a forest g over Σ, we denote
by f [⋆ 7→ g] the forest obtained by replacing every node labeled ⋆ in f with the
forest g. Similarly, given a context-template f over Σ ∪ {⋆} and a context c over
Σ, we denote by f [⋆ 7→ c] the forest obtained by replacing every subtree starting
at a node labeled ⋆ in f with the forest obtained by composing the context c
with the (possibly empty) forest given by all the subtrees rooted at a children
of the considered node labeled ⋆. Recall that we require contexts to be guarded,
so that f [⋆ 7→ c] is indeed a well-defined forest. In the special case where f is
the infinite unary tree labeled by ⋆, then for every context c, f [⋆ 7→ c] will be
denoted also by c∞ .
We call a forest-template guarded if no root is labelled by ⋆. Analogously
for context-templates. Let L ⊆ TΣc be a tree language over an alphabet Σ. We
4.2. PRELIMINARIES
95
say that two contexts, c1 , c2 , over Σ are L-equivalent (denoted by c1 ≡L c2 ) if
for every guarded context-template t over alphabet Σ ∪ {⋆}, t[⋆ 7→ c1 ] ∈ L if
and only if t[⋆ 7→ c2 ] ∈ L. Similarly, two forests f1 , f2 are L-equivalent (denoted
by f1 ≡L f2 ) if for every guarded forest-template t over alphabet Σ ∪ {⋆},
t[⋆ 7→ f1 ] ∈ L if and only if t[⋆ 7→ f2 ] ∈ L.
Note that since we are working with tree languages, we can safely ignore
forests t[⋆ 7→ f ] or t[⋆ 7→ c] that are not trees. Moreover, note that two trees
(seen as forests) can be L-equivalent while they do not agree on membership in
L.
It is easy to verify that:
Proposition 4.3. For every tree language L, L-equivalence is an equivalence
relation (both on contexts and on forests).
There is a natural generalization of forest-templates and context-templates,
where the holes are marked. More precisely, multi-forest-templates and multicontext-templates for an alphabet Σ are forests over an extended alphabet
Σ ∪ {⋆1 , . . . , ⋆n } (where n is a natural number), satisfying the same conditions as forest-templates and context-templates. Given a multi-forest-template
t over alphabet Σ ∪ {⋆1 , . . . , ⋆n } and forests f1 , . . . , fn , we denote by t[⋆1 7→
f1 , . . . , ⋆n 7→ fn ] the forest obtained by replacing every node labeled ⋆i in t with
the forest fi , with i = 1, . . . , n. Similarly for multi-context-templates.
The next substitution lemma for L-equivalence and multi-forest-templates
will be very useful:
Lemma 4.4. Let L be any MSO-definable Σ-tree language and f any multiforest-template over the extended alphabet Σ ∪ {⋆1 , . . . , ⋆n }. Consider a pair of
finite sequences of Σ-forests (g1 , . . . , gn ) and (h1 , . . . , hn ) such that gi and hi
are L-equivalent, for each i = 1, . . . , n. Then f [⋆1 7→ g1 , . . . , ⋆n 7→ gn ] and
f [⋆1 7→ h1 , . . . , ⋆n 7→ hn ] are also L-equivalent.
Proof. The proof is by induction on n. Let e be any guarded forest-template
over alphabet Σ ∪ {⋆}. For the base case we reason as follows. Because g ≡L h,
It follows that (e[⋆ 7→ f ])[⋆ 7→ g] ∈ L iff (e[⋆ 7→ f ])[⋆ 7→ h] ∈ L. Because (e[⋆ 7→
f ])[⋆ 7→ g] = e[⋆ 7→ (f [⋆ 7→ g])] and (e[⋆ 7→ f ])[⋆ 7→ h] = e[⋆ 7→ (f [⋆ 7→ h])] we
obtain that f [⋆ 7→ g] ≡L f [⋆ 7→ h].
For the induction step, let f be a multi-forest-template over the extended
alphabet Σ ∪ {⋆1 , . . . , ⋆n+1 } and (g1 , . . . , gn+1 ) and (h1 , . . . , hn+1 ) be a pair
of finite sequences of forests such that gi and hi are L-equivalent, for each
i = 1, . . . , n + 1. Denote by egn , resp. enh , the forest-template over Σ ∪ {⋆n+1 }
obtained by replacing every occurrence in e[⋆ 7→ f ] of a node labeled ⋆i by gi ,
resp. by hi , for i = 1, . . . n and denote by egn+1 , resp. ehn+1 , the multi-foresttemplate over the extended alphabet Σ ∪ {⋆1 , . . . , ⋆n } obtained by replacing
every occurrence in e[⋆ 7→ f ] of a node labeled ⋆n+1 by gn+1 , resp. by hn+1 .
First remark that, since the tree language L is MSO definable and therefore
closed under reordering of siblings, the following holds for every guarded foresttemplate e′ , every multi-forest template f ′ over Σ ∪ {⋆1 , . . . , ⋆n+1 } and every
′
sequence of Σ-forests (g1′ , . . . , gn+1
):
96
CHAPTER 4.
′
]∈L
eg′n′ [⋆n+1 7→ gn+1
⇔
′
e′ [⋆ 7→ (f ′ [⋆1 7→ g1′ , . . . , ⋆n+1 7→ gn+1
])] ∈ L
⇔
[⋆1 7→ g1′ , . . . , ⋆n 7→ gn′ ] ∈ L
e′n+1
g′
Thence, we have
e[⋆ 7→ (f [⋆1 7→ g1 , . . . , ⋆n+1 7→ gn+1 ])] ∈ L
⇔
egn [⋆n+1 7→ gn+1 ] ∈ L
⇔
(gn+1 ≡L hn+1 )
eng [⋆n+1 7→ hn+1 ] ∈ L
⇔
ehn+1 [⋆1 7→ g1 , . . . , ⋆n 7→ gn ] ∈ L
en+1
[⋆1 7→ h1 , . . . , ⋆n 7→ hn ] ∈ L
h
⇔
(induction hypothesis)
⇔
e[⋆ 7→ (f [⋆1 7→ h1 , . . . , ⋆n+1 7→ hn+1 ])] ∈ L
This shows that f [⋆1 7→ g1 , . . . , ⋆n+1 7→ gn+1 ] and f [⋆1 7→ h1 , . . . , ⋆n+1 7→ hn+1 ]
are also L-equivalent.
The previous proposition can analogously be proved to hold for multi-context
templates, but since we do not need it, we do not prove it.
From now on, when speaking about forest, resp. context, templates we
always mean guarded forest, resp. context, templates.
4.2.2
More on monadic second order logics
For n ≥ 1, we denote by ≡n denote the equivalence relation that holds between
two trees seen as relational structures if they cannot be distinguished by an
MSO-sentence of quantifier depth n (where both first-order and second-order
quantifiers are taken to contribute to the quantifier depth of a formula, i.e.,
qd(∃X.φ) = qd(∃x.φ) = qd(φ) + 1). It is well known that, for each n ≥ 1,
over a finite signature, there are only finitely many ≡n -classes of structures
(this can be shown by a straightforward induction, using also the fact that
a sentence of quantifier depth n cannot contain more than n first-order and
second-order variables). It follows that every structure can be completely described up to ≡n -equivalence by a single MSO-sentence of quantifier depth n.
Furthermore, the equivalence relation ≡n can be characterized using a variant
of Ehrenfeucht-Fraı̈ssé games. These games are defined as follows. Let M0 and
M1 be two relational structures over the same signature τ and n ∈ ω. The nround Ehrenfeucht-Fraı̈ssé game Gn (M0 , M1 ) is played by two players, Spoiler
and the Duplicator. Each player has to make n moves in the course of a play.
The players take turns. In his k-th move, Spoiler first selects a structure, M0
or M1 , and the kind of moves he want to perform. There are two possibilities:
either he chooses an element of the domain of the chosen structure, or one of
its subsets. If Spoiler picks an element of the domain of Mi , then Duplicator
chooses an element of the domain of M1−i , and if Spoiler selects a subset of the
4.2. PRELIMINARIES
97
domain of Mi , then Duplicator also chooses a subset of the domain of M1−i . At
the end of the play, we have obtained two sequences (a1 , . . . , ak , A1 , . . . , Al ) and
(b1 , . . . , bk , B1 , . . . , Bl ), where n = k + l, each ai is an element of the domain
of M0 and each Ai is a subset of the domain of M0 , each bi is an element of
the domain of M1 and each Bi is a subset of the domain of M1 . In order for
Duplicator to win the game, the resulting mapping (a1 , . . . , ak ) 7→ (b1 , . . . , bk )
should not only be a partial isomorphism with respect to the relations in the
structures, but should also preserve membership in the chosen sets A1 , . . . Al
and B1 , . . . Bl . It is then possible to prove that Duplicator has a winning strategy in the n-round game Gn (M0 , M1 ) (n ≥ 1) if and only if the two structures
M0 and M1 satisfy the same MSO-formulae of quantifier depth at most n. For
more details, see [52, 83].
Proposition 4.5. If L is an MSO-definable tree language, then there are only
finitely many L-equivalence classes of forests and contexts. Moreover, every Lequivalence class (of forests and of contexts) has a finitely branching and regular
member.
Proof. It is enough to verify that the proposition holds when considering only
non empty forests. We first show that there are only finitely many L-equivalence
classes (of non empty forests and of contexts) and that each is MSO-definable
(to make sense of this statement, we view non empty forests and contexts also
as relational structures, using an extra predicate P2 to denote the hole of a
context). Recall that for each n ≥ 1, over a finite signature, there are only
finitely many ≡n -classes of structures. Therefore, it is enough to show that nequivalence implies L-equivalence for large enough n. We will pick n simply to be
the quantifier depth of the MSO-formula that defines L. An easy EhrenfeuchtFraı̈ssé game-argument shows that for all forests f1 , f2 (viewed as relational
structures), if f1 ≡n f2 , then for every forest-template t, t[⋆ 7→ f1 ] ≡n t[⋆ 7→ f2 ],
hence t[⋆ 7→ f1 ] |= φ if and only if t[⋆ 7→ f2 ] |= φ, hence, by definition, f1 ≡L f2 .
Every element of t[⋆ 7→ f1 ] can be naturally identified either with an non-⋆
element of t, or with a pair consisting of an ⋆-element of t and an element of
f1 . Similarly for t[⋆ 7→ f2 ]. Duplicator’s strategy simply copies elements of
the first type, and for elements of the second type, copies the first coordinate
and applies his known winning strategy to the second coordinate of the pair. A
similar argument applies to contexts.
Next, note that every non-empty MSO-definable tree language contains a
finitely branching and regular tree. This follows from the automata theoretic
characterization of MSO given by Walukiewicz [130]. The same holds for MSOdefinable classes of forests (just by considering the tree language consisting of
all trees for which the forests consisting of all nodes except the root belongs
to the given forest language). Since we just saw that every L-equivalence class
of non empty forests is itself an MSO-definable class, we get the same for any
L-equivalence class of non empty forests and, similarly, of contexts.
Remark 4.6. It can be naturally asked what is the relation between the introduced notion of L-equivalence and the one originally introduced by Bojanczyk
and Idziaszek. It follows by the previous Proposition 4.5 that if L-equivalence
were defined using regular forest-templates and regular context-templates only,
the result would be the same as L-equivalence the way we defined, assuming that
98
CHAPTER 4.
L is a MSO definable tree language. This means that our definition essentially
coincides with the one used in [24] by the authors.
We still need a slightly more fine grained version of Proposition 4.5. Let c
be a context and f a forest. By a forest built from c and f , we will mean a
forest g for which there exists a forest s such that replacing each non-leaf node
in s by a copy of c and replacing each leaf-node of s by a copy of f yields g. We
then say that s is the skeleton of g.
Proposition 4.7. Let L be any MSO-definable tree language. Let c be a context
and f a forest. Every forest g built from c and f is L-equivalent to a forest g ′
built from c and f , whose skeleton is regular and finitely branching. Moreover,
if the skeleton of g satisfies some MSO-sentence ψ, then g ′ can be chosen so
that its skeleton satisfies ψ as well.
Proof. If g is the empty forest, the claim is easily verified. For the other case,
we reason as follows. First, we introduce some terminology. Let g be a forest
built from a context c and a forest f , whose skeleton is s. We can think of g
has a pair (s, hs ), where hs is a function assigning to each internal node of s the
context c and to each leaf of s the forest f . Given a node x of the skeleton s of
g, we call it a c-node or f -node, respectively, if hs (x) = c or hs (x) = f .
• By a region of g we mean the set of points that correspond to some point
of s. In other words, a region of g is some copy of f or of c (without the
hole) occurring in g.
• Consider another forest g ′ built from c and f , whose skeleton is s′ . We
can think of a point move of a player in the n-round Ehrenfeucht-Fraı̈ssé
game Gn (g, g ′ ) as the player placing a pebble on a node of the chosen forest, and a set move as the player placing pebbles on a set of nodes of the
chosen forest. So, by a pebbling of a tree, we mean a partial assignment
of nodes or sets of nodes to sets of pebbles. This means that a configuration of the game Gn (g, g ′ ) after round k can be described by a pair
((g, (P1 , . . . , Pk ), (g ′ , (P1′ , . . . , Pk′ )), where, for every i ≤ k, Pi and Pi′ are
either a set of nodes or a single node, and correspond to the positions of
the new pebbles introduced at round i.
Given a forest f , we say that two pebblings (P1 , . . . , Pk ) and (P1′ , . . . , Pk′ ′ ) of
f have same type if k = k ′ and Pi is a set of nodes iff Pi′ is a set of nodes. We say
that two pebblings of the same type of f are ≡n -equivalent, if Duplicator wins
the Ehrenfeucht-Fraı̈ssé game of length n between f with the first pebbling and
f with the second pebbling. Recall that for every type there are only finitely
many equivalence classes of this equivalence relation. Similarly for contexts c.
Let n be the quantifier depth of the MSO-formula defining L, Nf (k) be the
finite number of all ≡(n−(k−1)) -equivalence classes of types of pebblings of f
of length k ≤ n, and Nc (k) be the finite number of all ≡(n−(k−1)) -equivalence
P
classes of types of pebblings of c of length k ≤ n. Let ξk = ki=1 (Nf (i) +
Nc (i)) + 1) · 3i . Assume s satisfies the formula ψ. Let φs be an MSO formula
describing s completely, up to ≡ξn -equivalence. By Proposition 4.5, there is a
finitely branching regular forest s′ satisfying φs ∧ ψ. In particular, s ≡ξn s′ .
Let g ′ be the forest built from c and f whose skeleton is s′ . This concludes the
4.2. PRELIMINARIES
99
construction of g ′ . In the remainder of the proof we show that g ≡n g ′ , and
hence g ≡L g ′ .
Let’s describe the main idea of the proof. While playing the n-round game
Gn (g, g ′ ), Duplicator is maintaining on a scratch paper a ξn -round game on
Gξn (s, s′ ) where she is applying her existing winning strategy σ. Every time
Spoiler plays a move in Gn (g, g ′ ), say he plays something in g, then Duplicator
takes the pebbling of g, and “projects” it onto the individual regions of g. In
other words, Duplicator turns the pebbling of g into a function that assigns to
each node of s a pebbling of f or of c (namely the intersection of the pebbling of
g with this particular region). This means that after Spoiler’s move at round k,
every node x of s can be seen as being labeled by a sequence (Pk1 , . . . , Pkj ), and
we may refer to it as a pair (x, (Pk1 , . . . , Pkj )), where, whenever the sequence
(Pk1 , . . . , Pkj ) is not empty, each Pkl is either a set of nodes or a node of the
region hs (x), and the index kl ≤ k indicate at which round the new pebbles
corresponding to Pil have been introduced over hs (x). We call the sequence
(k1 , . . . , kj ) the projected type of the projected pebbling of g onto the region
hs (x) at round k. Thus, a player move at round k can be identified either with
a pair or with a set of pairs (x, (Pk1 , . . . , Pkj )) where kj = k. Now, in order to
determine her answer at round k, we want that Duplicator plays by standing
with a certain set of rules. This rules specifies a certain class of “well-behaved”
strategies for Duplicator and are as follows.
Well-behaved strategies : We distinguish between two cases depending on
whether Spoiler’s move in Gn (g, g ′ ) is a node or a set move. Assume that Spoiler
chooses g at round k, the case when Spoiler chooses g ′ being (mutatis mutandis)
the same:
(1) if Spoiler’s move is a node move characterized by (x, Pk ), where x is a
node of s and Pk is a node of hs (x), then:
i. Duplicator obtains the node σ(x) of s′ by playing x as a new Spoiler’s
move in the game Gξn (s, s′ ) on her scratch paper and then applying
the winning strategy σ to this move (notice that hs (x) = hs′ (σ(x)));
ii. then she chooses a node Pk′ in hs′ (σ(x)) and answers with the node
of g ′ corresponding to (σ(x), Pk′ );
iii. after playing her answer, Duplicator marks on her scratch paper the
new round played in the game Gξn (s, s′ ), round k in the game Gn (g, g ′ )
and the relation Rk = {((x, Pk ), (σ(x), Pk′ )}. Intuitively Rk keeps
trace of the relation between the new pebbling of g and the new pebbling of g ′ after round k.
(2) if Spoiler’s move at round k is a set move characterized by {(xl , Pkl ) : l ∈
Lk }, where Lk is the index set of the set and Pkl is a set of nodes of hs (xl ),
for each l ∈ Lk , then Duplicator’s move is obtained as follows:
• first the set is partitioned into two sets depending on whether a node
x is a c-node or a f -node,
• then, each set is partitioned by considering the projected type of
each pair, and each set corresponding of a certain projected type is
partitioned into ≡(n−(k−1)) -equivalence classes.
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CHAPTER 4.
Suppose that the partition is A1k , . . . , Aik . Each set Alk determines a set
A′lk of nodes of s. Notice that A′1k , . . . , A′ik are also pairwise disjoints.
Then:
i. Duplicator determines the pairwise disjoint subsets σ(A′1k ), . . . , σ(A′ik )
through her support game Gξn (s, s′ ) and her winning strategy σ by
playing ik new set moves.
ii. Thus, for every lk ≤ ik , and every node x′ ∈ σ(A′lk ), Duplicator
defines the pair (x′ , Pi′ ), where Pi′ is a set of nodes of hs′ (x′ ). Thence,
Duplicator’s answer is determined by the union of all new pebblings
over the obtained regions.
iii. After playing her answer, Duplicator marks on her scratch paper the
new ik rounds in the game Gξn (s, s′ ), Sthe new round in the game
Gn (g, g ′ ) and the new relation Rk = lk ≤ik {(x, Pk ) : x ∈ Alk } ×
{(x′ , Pk′ ) : n′ ∈ σ(Alk )}. In this case too, the relation Rk keeps trace
of the relation between the new pebbling of g and the new pebbling
of g ′ introduced at round k.
Suppose that Duplicator applies a strategy satisfying the previous rules.
Then, every configuration after round k can be seen as a triple
Ck = (C1k , C2k , (R1 , . . . , Rk ))
where:
(1) C1k is the actual configuration of the game Gn (g, g ′ ) after round k,
(2) C2k is the actual configuration of the game Gξn (s, s′ ) on her scratch paper
after i1 + · · · + ik rounds (where each ij depends on Spoiler’s move in
Gn (g, g ′ ) at round j),
(3) (R1 , . . . , Rk ) is the sequence of relations written on her scratch paper.
Remark that if Duplicator can reach a configuration Ck , she can survive for
at least ξn−(k−1) rounds from the actual configuration of the game Gξn (s, s′ )
written on her scratch paper. This means that for every k < n, if the game in
Gn (g, g ′ ) after round k is in the configuration Ck , then Duplicator can answer
to each move of her opponent by applying a well-behaved strategy and thence
reach a configuration Ck+1 .
Good configurations : Now, we want to show that there is a well-behaved
strategy that is winning. In order to do so, we first define a notion of a good
configuration. Intuitively a good configuration Ck implies that if the game on
g and g ′ had to stop in the corresponding current configuration, then it would
be winning for Duplicator, but it is stronger. More precisely, we show that in
every good configuration, Duplicator has a well-behaved strategy that ensures
that the configurations will always stay good. Because well-behaved strategies
preserving good configurations are winning for Duplicator in the game Gn (g, g ′ ),
we obtain the desired conclusion.
We say that a configuration Ck is good if:
(*) for every j ≤ k, if ((x, Pj ), (x′ , Pj′ )) ∈ Rj then:
4.2. PRELIMINARIES
101
• the projected pebbling of hs (x) and the projected pebbling of hs′ (x′ )
have same projected type and are ≡(n−(j−1)) -equivalent, and
• if (Pl′1 , . . . , Pl′y , Pj ) is the projected pebbling of hs (x), and (Pl′1 , . . . , Pl′y , Pj )
is the projected pebbling of hs′ (x′ ), then for every lz ∈ {l1 , . . . , ly },
((x, Plz ), (x′ , Pl′z )) ∈ Rlz .
Notice that if Ck is good, Cj is good for every j ≤ k. Now, suppose that
Duplicator applies a well-behaved strategy and reaches a good configuration Ck .
Then, because the strategy σ she is applying in the support game Gξn (s, s′ )
written on her scratch paper is winning and because for every j ≤ k and every
pair ((x, Pj ), (x′ , Pj′ )) ∈ Rj the projected pebbling of hs (x) and the projected
pebbling of hs′ (x′ ) are ≡(n−(k−1)) -equivalent, the mapping resulting from the
pebbling of g and the pebbling of g ′ after round k is a partial isomorphism
with respect to the relations in the structures which preserves membership in
the chosen sets. Trivially, the initial configuration (the configuration before
the game Gn (g, g ′ ) starts) is good. Thence, if we verify that from any good
configuration Ck Duplicator can apply a well-behaved strategy in such a way
that Ck+1 is also good, we have shown that Duplicator has a winning strategy
in Gn (g, g ′ ).
A well-behaved strategy preserving good configurations : Assume that
Spoiler chooses g at round k + 1 ≤ n, the case of g ′ being identical. We have
two cases to consider, depending on whether Spoiler’s move is either a node or
a set move.
(1) Assume that Spoiler’s move is a node move characterized by (x, Pk+1 ),
where x is a node of s, Pk+1 is a node of hs (x), and (Pl1 , . . . , Ply , Pk+1 ) is
the projected pebbling of the node hs (x) after his move. Then Duplicator
′
on σ(x) as follows. If the projected pebobtains the new pebbling Pk+1
bling of hs (x) is just (Pk+1 ), this means that hs (x) have no pebble on it
′
′
before this round. Thus, Duplicator defines the pair (x′ , Pk+1
), where Pk+1
equals Pk+1 . Otherwise, assume that (Pl1 , . . . , Ply ) is not empty. Because
Ck is good and σ is a winning strategy, there is a pebbling (Pl′1 , . . . , Pl′y )
of hs′ (σ(x)) such that:
• ((x, Plz ), (x′ , Pl′z )) ∈ Rlz for every lz ∈ {l1 , . . . , ly },
• the pebbling (Pl1 , . . . , Ply ) of hs (x) and the pebbling (Pl′1 , . . . , Pl′y ) of
hs′ (σ(x)) have same type and are ≡n−(k−1) -equivalent.
′
This means that there is a set of nodes Pk+1
such that the pebbling
′
(Pl1 , . . . , Ply , Pk+1 ) of hs (x) and the pebbling (Pl′1 , . . . , Pl′y , Pk+1
) of hs′ (σ(x))
′
are ≡(n−k) -equivalent. Thus Duplicator answers with the node (σ(x), Pk+1
).
l
(2) Suppose that Spoiler’s move is characterized by {(xl , Pk+1
) : l ∈ Lk+1 },
l
where Pk+1 is a set of nodes of h(xl ), for each l ∈ Lk+1 . Let σ(A′1k+1 ), . . . ,
σ(A′jk+1 ) be the pairwise disjoint sets obtained by Duplicator by playing
jk+1 new set moves in her support game Gξn (s, s′ ). Thus, for every lk+1 ≤
jk+1 , and every node x′ ∈ σ(A′lk+1 ), Duplicator determines the set of nodes
′
Pk+1
associated to hs′ (x′ ) as follows. Let x be any element of A′lk+1 .
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CHAPTER 4.
• If the projected pebbling of hs (x) is just (Pk+1 ), this means that hs (x)
have no pebble on it before this round. Thus, Duplicator defines the
′
′
pair (x′ , Pk+1
), where Pk+1
is a set of nodes of hs′ (x′ ) such that
′
′
hs′ (x ) with pebbling Pk+1 is ≡(n−k) -equivalent with every member
of Alk+1 .
• Otherwise, assume that (Pl1 , . . . , Ply , Pk+1 ) is the projected pebbling
of the node hs (x). Because Ck is good and σ is a winning strategy,
there is a pebbling (Pl′1 , . . . , Pl′y ) of hs′ (x′ ) such that:
– ((x, Plz ), (x′ , Pl′z )) ∈ Rlz for every lz ∈ {l1 , . . . , ly },
– the pebbling (Pl1 , . . . , Ply ) of hs (x) and the pebbling (Pl′1 , . . . , Pl′y )
of hs′ (x′ ) have same type and are ≡(n−(k−1)) -equivalent.
′
This implies that there is a set of nodes Pk+1
such that the pebbling
′
(Pl1 , . . . , Ply , Pk+1 ) of hs (x) and the pebbling (Pl′1 , . . . , Pl′y , Pk+1
) of
hs′ (x′ ) are ≡(n−k) -equivalent, meaning that the projected pebbling
of hs′ (x′ ) is in the same ≡(n−k) -equivalence class as the projected
pebbling of each member of A′lk+1 .
In both cases the obtained configurations clearly satisfy condition (∗).
4.2.3
The logic EF and EF-bisimulation
Fix an alphabet Σ. The set of formulae of EF over Σ is defined by the grammar
φ ::= a | φ ∧ φ | ¬φ | EFφ
(a ∈ Σ)
The semantics of EF over non empty trees is defined inductively as usual by
saying that every EF-formula a ∈ Σ is true in trees with root label a and that
an EF formula EFφ is true in trees that have a proper subtree where φ is true.
For any EF formula φ, the class of trees where φ is true is denoted by L(φ).
Given a tree language L, we say that L is EF-definable if there is an EF-formula
φ such that L = L(φ). It is well-known that (see for example [31]):
Proposition 4.8. Every EF-definable tree language is also WMSO-definable
(and, in fact definable in first-order logic with the descendant relation).
Following [24], we introduce a special bisimilarity game on forest, called the
EF bisimulation game. We first define the game in the case of trees. Let t0
and t1 be two trees. The EF bisimulation game over t0 and t1 is played by two
players: Bob and Anne. The game proceeds in rounds. At the beginning of
each round, the state in the game is a pair of trees (t′0 , t′1 ). A round is played
as follows. First if the root labels a0 , a1 of t′0 , t′1 are different, then Bob wins
the whole game. Otherwise Bob selects one of the trees t′i , for i = 0, 1, and its
subtree si . Then Anne selects a subtree s1−i in the other tree t′1−i . The round
is finished, and a new round is played with the state updated to (s0 , s1 ). If
Anne can survive for infinitely many rounds in the EF bisimulation game on t0
and t1 , then we say that the trees t0 and t1 are EF-bisimilar.
Note that clearly if two trees are bisimilar in the standard way, they also
are EF-bisimilar. The converse need not to be true. Consider for example the
tree t on the alphabet {a, b} where the only nodes labelled by a are the nodes
02k+1 , with k > 0, and the tree t′ on the alphabet {a, b} where the only nodes
4.2. PRELIMINARIES
103
labelled by a are the nodes 02k , with k > 0. The two trees are EF-bisimilar but
not bisimilar.
A tree language L is called invariant, or closed, under EF-bisimulation if it
is impossible to find two trees, t0 ∈ L and t1 ∈
/ L that are EF-bisimilar. From
the previous remark on the interrelation between standard bisimilarity and EF
bisimilarity, if a tree language is invariant under EF-bisimulation, then it is also
invariant under standard bisimulation, but the converse is in general not true.
This game is so designed that all tree languages defined by an EF formula
are invariant under EF-bisimulation. Formally, we have that:
Proposition 4.9 ([24]). Every EF-definable tree language is invariant under
EF-bisimulation.
The converse is not true. The typical counter-example is the language of all
finite trees over a fixed finite alphabet. This language is invariant under EFbisimulation but it cannot be defined by an EF-formula, as follows from a nice
result of Bojanczyk and Idziaszek. Let us say that a context c is L-idempotent
if the composition c(c) is L-equivalent to c.
Theorem 4.10. A MSO definable tree language L can be defined by an EF
formula if and only if
1. L is invariant under EF-bisimulation;
2. for every L-idempotent context c and for every forest f , c(f ) ≡L (c + c(f ))∞
Moreover the previous two conditions are decidable.
Remark 4.11. This characterization of the logic EF was proved by Bojanczyk
and Idziaszek in [24] for the case of finitely branching trees. However, it follows
from Proposition 4.5 that the result holds for arbitrarily branching trees as
well. Also, strictly speaking, the result in [24] is different as it characterizes
definability by EF-formulae that are Boolean combinations of formulae of the
form EFφ. However, as explained in [24], this is not an essential restriction. More
precisely, call a EF-formula ψ a EF-forest formula if ψ is a Boolean combinations
of formulae of the formVEFφ. Then, one can prove that every EF-formula φ is
logically equivalent to a∈A (a → φa ), where each φa is a EF-forest formula.
This means that a tree language L is EF-definable iff, for every a ∈ A, the
language La is definable by an EF-forest formula, where a tree t is said to be in
La iff the tree obtained from t by relabeling its root with a is in L. Thus, the
equivalence relation ≡L in the previous theorem is, strictly speaking, given by
the finite intersection of all ≡La . It is easily checked that all relevant properties
of L-equivalence that we use in the proof of the main theorem still hold while
thinking of this equivalence relation as given by the previous finite intersection.
We extend the notion of EF-bisimularity to forests by saying that two forests
f1 , f2 are EF-bisimilar if the trees obtained from f1 and f2 by adding a “fresh”
root, are EF-bisimilar. More precisely, let f1 and f2 be two forests over Σ. Let
t1 and t2 be any pair of trees over Σ ∪ {a}, a ∈
/ Σ, such that: (i) each forest fi
is obtained from the corresponding tree ti by removing the root node and (ii)
t1 (ε) = t2 (ε) = a. Then we say that f1 and f2 are EF-bisimilar if the trees t1
and t2 are EF-bisimilar. Note that for forests f1 and f2 consisting of a single
104
CHAPTER 4.
tree t1 and t2 respectively, saying that f1 and f2 are EF-bisimilar is not the
same as saying that t1 and t2 are EF-bisimilar.
The following lemma, relating EF-bisimilarity to L-equivalence, will come
in handy later on. Call two contexts EF-bisimilar, if they are bisimilar when
viewed as forests over an alphabet containing an additional label “2”.
Lemma 4.12. Let L be any EF-bisimulation-invariant tree language. Then every
two EF-bisimilar forests are L-equivalent and every two EF-bisimilar contexts
are L-equivalent.
Proof. It follows from the fact that if f1 and f2 are EF-bisimilar, then for every
(guarded) forest-template t (with a single root), t[⋆ 7→ f1 ] is EF-bisimilar to
t[⋆ 7→ f2 ], and similarly in the case of contexts.
4.3
The complexity of conciliatory tree languages
Analogously to the space of full trees, the topology we will consider here on
conciliatory trees is the prefix topology, where the open sets are, intuitively,
those sets of trees for which membership of a tree is witnessed by a finite-depth
prefix of the tree. Thus, for example the set of trees containing a node labeled
a is an open set, but the complement is not.
We need first an analogous of the notion of initial tree but for the conciliatory
case. For a conciliatory tree t over Σ and a natural number n ≥ 1, the depth-n
prefix of t, denoted by t(n) , is the Σ-tree obtained by restricting the domain of
t to the finite sequences of natural numbers of length at most n − 1. We say
that two trees t, t′ are equivalent up to depth n if t(n) = t′(n) , i.e., for all finite
sequence w of natural numbers of length at most n − 1, (i) w ∈ dom(t) if and
only if w ∈ dom(t′ ), and (ii) t(w) = t′ (w) if t and t′ are defined on w. We call
a set X of conciliatory Σ-trees open if for each t ∈ X there is a natural number
n ≥ 1 such that for all conciliatory trees t′ over Σ, if t and t′ are equivalent up
to depth n then t′ ∈ X. This indeed yields a topological space. When we say
that a set of trees is Borel, we will mean that it is Borel with respect to this
topology.
Theorem 4.13. Every WMSO-definable set of trees is Borel.
Proof. The proof is by induction on the size of the WMSO-formula. For the
induction, it is convenient to prove the following, slightly stronger, result: for
each WMSO formula φ and valuation λ for the free variables of φ, the set of
trees that are consistent with λ and satisfy φ under valuation λ is Borel.
First, observe that if λ is a valuation for a finite set of variables, then the
class of trees that are consistent with λ is an open set. Indeed, if n is the depth
of the longest element in the image of λ, then whether a tree is compatible with
λ is determined by the depth-n prefix of the tree.
Next, the proof is by induction on the structure of φ. If φ is an atomic
formula of the form x < y, x = y, or X(x), then for all valuations λ, the set of
trees consistent with λ and satisfying φ under valuation λ is either the empty
set or the entire space, both of which are open sets. If φ is an atomic formula of
the form Pa (x), the truth of φ in a tree is determined by any prefix of the tree
containing the node x, and hence, for every valuation λ, the set of trees consistent with λ and satisfying φ under valuation λ is an open (in fact, clopen) set.
4.3. THE COMPLEXITY OF CONCILIATORY TREE LANGUAGES
105
The induction step for conjunction, disjunction, negation, first-order existential
quantification, and weak second-order existential quantification, is straightforward, given that the number of elements of N∗ , as well as the number of finite
subsets of N∗ , is countable.
In the following subsection, we prove that the following is an example of a
tree language that is not Borel. Recall that a tree is well-founded if it has no
infinite branch.
Theorem 4.14. The set WF of all well-founded trees over the alphabet {a} is
c
not a Borel subset of the space T{a}
.
Remark 4.15. Usually the set WF is shown to be not Borel by identifying each
tree t over {a} with its characteristic function, and therefore thinking the space
c
T{a}
as a subset of the Cantor space 2ω , see for example [70]. However it is
not clear, at least for us, how to determine that this standard result of classical
descriptive set theory immediately implies that the set WF, see as a subset of
c
T{a}
and not of 2ω , is not Borel. This is the reason why in the next subsection
we present a proof for Theorem 4.14.
Remark 4.16. Traditionally, the study of the topological complexity of tree
language has focussed on full trees. Here, we are interested in arbitrary (conciliatory) trees. However, the two settings are not very different.
First of all, by Theorem 4.13, the set of full trees is itself a Borel set (with
respect to the space of conciliatory trees). It follows that a set of full trees is
Borel if and only if it is Borel within the subspace consisting of full trees only.
Secondly, we can associate in a natural way to each tree t over alphabet Σ
the full tree ts over alphabet Σ∪{s} (for some s 6∈ Σ) obtained by padding t with
s, i.e., for each w ∈ N∗ , ts (w) = t(w) if w ∈ dom(t), and ts (w) = s otherwise.
Conversely, every full tree t over alphabet Σ ∪ {s} naturally gives rise to a
(possibly empty) tree over the alphabet Σ, namely the subtree consisting of all
nodes that are not labeled s and do not have an ancestor labeled s. We call this
subtree the undressings of t. Now, for any tree language L over alphabet Σ, let
us define Ls to be the set of full trees t over alphabet Σ ∪ {s} whose undressings
belongs to L. By induction on the Borel rank, it can then be easily verified that
if a set of conciliatory trees L is Borel then the set of full trees Ls is also Borel.
The next proposition, determining a sufficient condition for a function in
order to be continuous, will be very useful.
Proposition 4.17. Let Σ and Σ′ be two finite sets, X be either TΣ or TΣc , Y
be either TΣ′ or TΣc ′ , and F be a function from X into Y . Then, if for every
depth-n prefix t(n) over Σ there exists a depth-n prefix t′(n) over Σ′ such that
F (t(n) · X) ⊆ t′(n) · Y , F is continuous.
Proof. Consider a set t′(n) · Y , where t′(n) is a depth-n prefix over Σ′ . If we show
that the counterimage of t′(n) ·Y is an open subset of X we are done. Let P be the
set of all depth-n prefix t(n) over Σ such that F (tS(n) ·X) ⊆ t′(n) ·Y . Then, on the
one hand because F is a function we have that t(n) ∈P t(n) · X ⊆ F −1 (t′(n) · Y ).
On the other hand by definition of P and the
of F , we have that
S property
S
(n)
(n)
−1 ′(n)
t
·
X = F −1 (t′(n) · Y ),
t
·
X
⊇
F
(t
·
Y
),
and
therefore
t(n) ∈P
t(n) ∈P
meaning that the counterimage of t′(n) · Y is an open set.
106
CHAPTER 4.
Note that the previous proposition clearly still holds if we consider X (or
Y ) as the topological space consisting of the set of all full, resp. conciliatory,
binary trees over Σ (or Σ′ ).
4.3.1
The set of well-founded trees is not Borel
In this subsection, we prove Theorem 4.14, i.e., we show that the set of wellfounded trees over a singleton alphabet {a}, denoted by WF, is not Borel. As
we explained in Remark 4.16, in order to show that a set of trees L is not Borel,
it is enough to show that Ls is not Borel. Now, it is not hard to see that WFs
consists of all full trees t : N∗ → {a, s} with the property that on every branch
there is at least a node labelled with s. We show that this set is not Borel.
We use the fact that, from Lemma 1.3, the set Wa of all full binary trees
over {a, b} where in every branch there are only finitely many nodes with label
a is not Borel. We will construct a continuous function F from full binary
trees over the alphabet {a, b} to full trees over the alphabet {a, s}, such that
F −1 (WFs ) = Wa . This shows that Wa ≤W WFs . Since Wa is not Borel, we can
then conclude that WFs (and hence also WF) is not Borel.
Intuitively, F (t) will be a tree whose branches correspond to sequences of
descendant-steps in t, possibly skipping over intermediate nodes, so that if t
contains a branch in which a appears infinitely often, F (t) contains an infinite
branch containing entirely of a-nodes, and vice versa. In this way, we have
t ∈ Wa iff F (t) ∈ WFs . We have to take care also that F is a continuous
function. It does not suffice to define F (t) simply as the tree obtained from
t by recursively adding, for each node w having a descendant v labeled a, an
extra copy of v as a direct child of w, because this would not yield a continuous
function. Instead, we employ a slightly more refined construction.
Let <bf be the breath-first order on {0, 1}∗, i.e., the ordering in which ǫ <
0 < 1 < 00 < 01 < 10 < 11 < 000 < . . .. Observe that this order has order type
ω, i.e., ({0, 1}∗, <bf ) is isomorphic to (N \ {0}, <). Let ι be the isomorphism
in question, i.e., ι(ǫ) = 1, ι(0) = 2, ι(1) = 3, ι(00) = 4, etc. Similarly, for each
x ∈ {0, 1}∗, let ιx be the isomorphism between (x · {0, 1}∗ , <bf ) and (N \ {0}, <)
mapping sequences starting with x bijectively onto the positive natural numbers
in an order preserving way. Intuitively, for all x, y ∈ {0, 1}∗, ιy (x) is the index
of the node x within the binary subtree rooted by y, following the breadth-first
ordering.
Now, define the function I : {0, 1}∗ → ℘(N∗ ) recursively on the length of all
finite words over {0, 1} as follows:
(1) I(ε) = {ι(ε)ι(ε) } = {1}
(2) I(x) = {ι(x)ι(x) } ∪ {vιy (x)ιy (x) : y ancestor of x, v ∈ I(y)}
Recall that given x ∈ N, for all n ∈ N, xn , is defined as: x0 = ε, xn+1 = xn x.
This means that, for each x, y ∈ {0, 1}∗ and v ∈ N∗ , ι(x)ι(x) = ι(x) · · · ι(x) and
{z
}
|
ι(x) times
ιy (x)
therefore vιy (x)
= v ιy (x) · · · ιy (x), where by definition ι(x), ιy (x) > 0.
|
{z
}
ιy (x) times
Finally, for any full binary {a, b}-tree t, we define F (t) to be the tree F (t) :
N∗ → {a, s} defined as follows:
4.4. CHARACTERIZATIONS OF EF OVER INFINITE TREES
(
107
if t(x) = a
,
if t(x) = b
S
S
(2) for every w ∈
/ x∈{0,1}∗ I(x) which is an ancestor of an element of x∈{0,1}∗ I(x),
F (t)(w) = a
∗
(1) for every x ∈ {0, 1} , and every v ∈ I(x), F (t)(v) =
a
s
(3) to all other nodes u ∈ N∗ , F (t)(u) = s
We verify that:
Claim 4.18. t ∈ Wa iff F (t) ∈ WFs .
Proof of the Claim : First note that dom(F (t)) = N∗ . If t ∈
/ Wa , then there
is a branch with infinitely many nodes labelled by a. Suppose those nodes,
enumerated consistently with the prefix order, are x0 , x1 , . . . . Every xi is an
ancestor of xi+1 in the full binary tree, meaning that
F (t)(ι(x1 )ι(x1 ) ) = a
F (t)(ι(x1 )ι(x1 ) ιx1 (x2 )ιx1 (x2 ) ) = a
F (t)(ι(x1 )ι(x1 ) ιx1 (x2 )ιx1 (x2 ) ιx2 (x3 )ιx2 (x3 ) ) = a
..
.
Thus there is an infinite branch in F (t) where all nodes are labelled by a, and
therefore F (t) ∈
/ WFs . For the other direction, suppose that F (t) ∈
/ WFs and
that there is a infinite branch π only labelled by a.SBy construction of F (t),
there are infinitely many nodes of π which are in x∈{0,1}∗ I(x). For every
S
n ∈ N, let π(n), π(n + 1) ∈ x∈{0,1}∗ I(x). Therefore there are x, y ∈ {0, 1}∗
such that π(n) ∈ I(x) and π(n + 1) ∈ I(y), meaning that t(x) = t(y) = a. By
construction of I, y is reachable from x, showing that t ∈
/ Wa . End of proof of
claim.
Claim 4.19. F is a continuous function.
Proof of claim: By construction of the function F , for every depth-n binary
prefix t(n) over {a, b} there exists a depth-n prefix t′(n) over {a, s} such that
F (t(n) · X) ⊆ t′(n) · T{a,s} , where X is the space of all full binary trees over
{a, b}. Thus F is continuous by Proposition 4.17.
End of proof of claim.
Hence Wa ≤W WFs and Wa is not Borel, meaning that WFs is not Borel
either.
4.4
Characterizations of EF over infinite trees
Everything now is ready to prove the equivalence of the given characterizations
of the logic EF on trees.
Theorem 4.1. Let L be any MSO-definable tree language. The following
conditions are equivalent and decidable:
(1) L is EF-definable
108
CHAPTER 4.
(2) L is WMSO-definable and closed under EF-bisimulation
(3) L is Borel and closed under EF-bisimulation
(4) L is closed under EF-bisimulation, and for every L-idempotent context c,
and for every forest f , c(f ) and (c + cf )∞ are L-equivalent.
Proof. We prove the equivalences in a round-robin fashion. The decidability
follows from Theorem 4.10.
(1)⇒(2): If a tree language L is EF-definable, by Proposition 4.9 we know
that L is invariant under EF-bisimulation and by Proposition 4.8 we know that
it is also WMSO-definable.
(2)⇒(3): This follows from Theorem 4.13.
(3)⇒(4): The proof is by contraposition. Suppose there is an idempotent
context c and a forest t such that c(t) and (c + c(t))∞ are not L-equivalent.
There are thence two possibilities:
(a) there is a (guarded) forest-template e such that
e[⋆ 7→ c(t)] ∈ L and e[⋆ 7→ (c + c(t))∞ ] 6∈ L,
(b) there is a (guarded) forest-template e such that
e[⋆ 7→ c(t)] 6∈ L and e[⋆ 7→ (c + c(t))∞ ] ∈ L.
We will show that both this two possibilities imply that L is not Borel, by giving
a continuous reduction of the non-Borel set WF or its complement to L. Recall
that WF is the set of all well-founded trees over an alphabet consisting of a
single letter a.
For any tree t′ over alphabet {a}, let tb′ be the forest obtained as follows:
if t′ is the empty tree then tb′ is ct, otherwise tb′ is obtained by replacing every
leaf node of t′ by the forest ct, and replacing every non-leaf node by the context
(c + ct). Finally, let F (t′ ) be the tree e(tb′ ), i.e., the result of replacing every ⋆
in e by tb′ . Because for every finite k, the first k levels of t determines the first
k levels of F (t), by Proposition 4.17 we know that F is continuous.
Claim 4.20.
(i) if t′ ∈ WF, then tb′ is L-equivalent to ct
(ii) if t′ ∈
/ WF, then tb′ is L-equivalent to (c + ct)∞
This implies that L is not Borel. Indeed, suppose that case (a) holds. From
the previous claim we obtain that t′ ∈ WF if and only if F (t′ ) ∈ L. This means
that F −1 (L) = WF, proving that WF ≤W L. Analogously, if case (b) holds,
from the previous claim we obtain that t′ ∈ WF∁ if and only if F (t′ ) ∈ L, where
WF∁ is the complement of WF. Thus F −1 (L) = WF∁ and therefore WF∁ ≤W L.
It remains only to prove the above claim.
We first need to introduce some terminology. Clearly, tb′ is a forest that is
built from the context c and the tree t. Let us denote the skeleton of tb′ by s.
Suppose that s is not well-founded. Let x any node in the domain of s such
that the subtree s.x is well-founded and there is no ancestor y of x such that
4.5. MODAL=BOREL, ON TRANSITIVE MODELS
109
s.y is also well-founded. Then we call the subforest of tb′ built up from c and t
whose skeleton is s.x a maximal well-founded subforest of tb′ .
We first prove claim (i). Suppose that t′ ∈ WF. Then, clearly, s is wellfounded. By Proposition 4.7, tb′ is L-equivalent to a finitely branching, regular
forest t′′ built up from c and t whose skeleton s′′ is well-founded (recall that
well-foundedness is MSO-definable). Since every infinite finitely branching tree
has an infinite branch, we then know that the skeleton s′′ is in fact finite. Let
k be the length of the longest branch of s′′ . It is easy to see that s′′ is EFbisimilar to a tree that has a single path whose labels (read from the root to the
leaf) form the word ck+1 t. For convenience, we denote this tree itself by ck+1 t.
Analogously, it is not hard to see that t′′ is EF-bisimilar to the forest built up
from c and t whose skeleton is ck+1 t, i.e., the forest ck+1 t. Since EF-bisimilarity
implies L-equivalence (Lemma 4.12), this means that tb′ is L-equivalent to ck+1 t.
And since c is an L-idempotent context, we have that tb′ is in fact L-equivalent
to ct.
Next, we prove claim (ii). Suppose that t′ 6∈ WF. As before, clearly, s is
non well-founded. By Proposition 4.7, tb′ is L-equivalent to a regular (finitelybranching) forest t′′ built up from c and t whose skeleton s′′ is non well-founded
(recall that non well-foundedness is MSO-definable). Because tb′′ is regular, there
are only finitely many maximal well-founded subforests of tb′′ built up from c and
t. Thus since we have already shown that every well-founded forest built from
c and t is L-equivalent to ct, we can take all maximal well-founded subforests
of tb′′ built up from c and t and replace each of these subforests by ct. Here, we
use the substitution lemma (Lemma 4.4). Furthermore, using the fact that EFbisimilarity implies L-equivalence (Lemma 4.12), if the resulting forest contains
several copies of ct next to each other as siblings, they can be collapsed into one.
This yields a new forest that can be viewed as a forest built up from the context
c + ct alone, whose skeleton has no leafs. But any such forest is EF-bisimilar,
hence L-equivalent, to the forest (c + ct)∞ .
(4)⇒(1): This is given by Theorem 4.10.
4.5
Modal=Borel, on transitive models
Everything now is ready to show that on transitive transition systems, modal
logic is the Borel fragment of the modal µ-calculus. Without loss of generality,
from now we suppose that Prop is a finite set. Since models of EF are trees,
we first have to relate transitive transition systems with trees. For a start,
recall that a tree t : N∗ → ℘(Prop) can be seen as a pointed transition system
(Tt , s), where S = dom(t), the initial state s is just the root ǫ of t, the transition
relation is the child relation, and the valuation λ : Prop → ℘(S) is just given by
λ(p) = {w ∈ dom(t) : p ∈ t(w)}.
Given a pointed transition system (T , s), its tree unraveling is the set T (T , s)
of trees t over ℘(Prop) such that (Tt , ǫ) is bisimilar with (T , s). In
S general, given
a set P of pointed transition systems, by T (P ) we denote the set (T ,s)∈P T (T , s).
In Subsection 2.2.2, we associated to every µ-formula ϕ, another µ-formula
ϕtr such that for every transition system T it holds that
s ∈ kϕtr kT
if and only if
s ∈ kϕkT tr ,
110
CHAPTER 4.
where T tr is the transitive closure of T (Lemma 2.4). This immediately leads
to:
Lemma 4.21. For every µ-formula ϕ,
{(T tr , s) : (T , s) ∈ kϕtr k} = kϕkt
Thus, we say that the set of transitive models of a µ-formula ϕ is Borel iff
the set of the tree unravelings of the models of the associated formula ϕtr is a
Borel set. We also say that over transitive models a µ-formula is modal iff there
is a modal formula ψ ∈ LM such that kψkt = kϕkt .
Before obtaining the desired characterization, we need a small lemma relating EF-bisimulation on trees with bisimulation on transitive models:
Lemma 4.22. Let ϕ be a µ-formula. Then the set T (kϕtr k) is closed under
EF-bisimulation.
Proof. We know that kϕktr and kϕtr k are closed under bisimulation. Thus
suppose that t′ is EF-bisimilar to a tree t ∈ T (kϕtr k). This implies that (Tt , ǫ) ∈
kϕtr k and by Lemma 4.21 that (Tttr , ǫ) ∈ kϕktr . But the winning strategy
for Anne in the EF game on t and t′ is also a winning strategy for her in the
tr
tr
tr
bisimulation game on (Tttr
′ , ǫ) and (Tt , ǫ). This means that (Tt′ , ǫ) and (Tt , ǫ)
tr
tr
are bisimilar and therefore that (Tt′ , ǫ) ∈ kϕk . By applying Lemma 2.4 we
obtain that t′ ∈ T (kϕtr k).
Finally, we have that:
Corollary 4.23. For every µ-formula ϕ, on transitive models ϕ is modal iff
kϕkt is Borel.
Proof. Clearly for every EF-formula φ, there is a modal formula ψ such that
for every tree t over ℘(Prop), φ is true over t iff (Tt , ǫ) ∈ kψtr k. Thus, if ϕ
is modal, by Theorem 4.1, T (kϕtr k) is a Borel set. For the converse, suppose
that kϕkt is Borel. By definition, this means that T (kϕtr k) is a Borel set. But
on the one hand we have that T (kϕtr k) is recognizable by a µ-automaton and
therefore MSO definable. On the other hand, by Lemma 4.22, this set is closed
under EF-bisimulation, and therefore once more by Theorem 4.1, T (kϕtr k) is
EF-definable, and therefore there is modal formula ψ such that kψkt = |ϕkt .
4.6
Summarizing remarks
Understanding the expressive power of logics over words or trees is an important
problem that can be found in many areas of Computer Science. Examples of
such logics are temporal logics for system verification but also tree navigation
languages for XML. If for (finite) words this type of problem is well-studied
and understood, the same cannot be said for trees, the situation being even
worse when considering infinite trees. A first result going in the direction of
filling this gap has been obtained recently by Bojanczyk and Idziaszek [24].
In their work, they give a nice effective algebraic characterization on infinite
finitely branching trees of the logic EF, a simple temporal logic which allows
us to express statements such as “whenever a request is made it is eventually
granted”. In this chapter we have used Bojanczyk and Idziaszek result by
4.6. SUMMARIZING REMARKS
111
providing several equivalent characterizations for this logic on arbitrary trees
(finite and infinite, finitely and infinitely branching). More specifically we have
proved that the properties of:
(1) being definable by an EF formula,
(2) being a Borel set and being closed under EF-bisimulation, and
(3) being definable by a formula of the weak monadic second order logic with
the child relation and being closed under EF-bisimulation,
all coincide for regular languages of both finite and infinite trees. Because
all the previous properties are proved to be equivalent to the characterization
given by Bojanczyk and Idziaszek for the finitely branching case, we also get
their decidability.
As an almost immediate corollary of the main result of the chapter, we have
finally obtained that over transitive models, modal logic is the Borel fragment
of the modal µ-calculus.
112
CHAPTER 4.
Part II
Hierarchical Questions for
Tree Languages Definable
Without Alternation
113
Chapter 5
Preliminaries
The Mostowski–Rabin (index) hierarchy, the Borel hierarchy, and the Wadge hierarchy are the most common measures of complexity of recognizable languages
of infinite words or trees.
We have seen that the first one orders languages according to the nesting of
positive and negative conditions checked by the recognizing automaton, and it
has two main versions: weak and strong.
The classical Borel hierarchy is based on the nesting of countable unions
and negations in the set theoretic definition of the language, starting from
the simplest (open) sets. It drew attention of automata theorists as early as
1960s [79], and has continued to inspire research efforts ever since, mainly because of its intimate relations with the index hierarchy [61, 101, 114]. Finally,
the Wadge hierarchy is the least known of the three. When restricted to Borel
sets, it is an almost ultimate refinement of the Borel hierarchy. Defined by
the preorder induced on languages by simple (continuous) reductions, it enables precise comparison of different models of computation by associating a
certain ordinal (height) to the Wadge hierarchy restricted to the classes under
comparison [49, 55, 92, 111].
Measuring hardness of recognizable languages of infinite trees is a long standing open problem. Unlike for infinite words, where the understanding is almost
complete since Wagner’s 1977 paper [126], for trees the only satisfyingly examined case is that of deterministic automata [90, 92, 93, 101, 102]. But the
deterministic and non-deterministic case differ immensely for trees. The only
results obtained for non-deterministic or alternating automata are strictness
theorems for various classes [32, 33, 89, 96], and lower bounds for the heights
of the hierarchies [51, 114]. To the best of our knowledge, the only nontrivial decidability results are the famous Rabin’s theorem for non-deterministic
tree automata, and decision procedures for low levels of the index hierarchy of
alternating tree automata [129, 78, 103].
This part of the dissertation intents to change this situation, even if only very
slightly for a start, by looking at the case of weakly recognizable languages, or
equivalently to tree languages definable by a formula of the bi-modal µ-calculus
without alternation [8, 96].
After introducing the projective hierarchy, the Borel Wadge hierarchy and
some further basic notions concerning automata, we briefly discuss the Wadge
hierarchy of weakly recognizable tree languages. Then in Chapter 6, we pro115
116
CHAPTER 5.
pose a subclass of weak alternating automata having all three hierarchies (weak
index, Borel and Wadge) decidable and capturing a reasonable amount of nondeterminism. The class we advocate, linear game automata (LGA), is obtained
by taking linear automata (a.k.a. very weak automata), that emerged in the
verification community, and restricting the alternation to the choice of a path in
the input tree. Linear automata capture CTL [76], which is expressive enough
for many applications. Though linear game automata are weaker, they retain
most alternation related to the branching structure. Evidence for their expressivity is topological: they recognize sets of arbitrarily high finite Borel rank,
and their Wadge hierarchy has the height (ω ω )ω , much larger than (ω ω )3 + 3
for deterministic automata.
5.1
Topological hierarchies
In classical descriptive set theory, definable subsets of Polish spaces are classified
according to the complexity of their definition in terms of projection, countable
unions and complementation. The results of this classification are the Borel and
projective hierarchies. A way to compare a pair of sets L and M is thence to
compare their respective positions in those hierarchies. There exists however a
more natural and refined complexity measure: the Wagde reduction, introduced
in subsection 1.2.2. This order induces a hierarchy of sets that refines immensely
the Borel hierarchy and, assuming a set-theoretic axiom of determinacy, also the
projective hierarchy. From this point of view, to compare any pair of sets L and
M is to compare their positions in the Wadge hierarchy, that is their Wadge
degrees.
We have already seen that the space TΣ of all full binary trees equipped with
the initial segment topology is a Polish space. This means that we can use tools
and methods from descriptive set theory in order to study the complexity of
classes of recognizable tree languages. In order to do so for weakly recognizable
languages, we first introduce the classical projective and Borel Wadge hierarchies1 . Then, as the final subsection of this preliminary chapter, we discuss the
Wadge hierarchy of tree languages recognized by weak alternating automata.
5.1.1
The topological complexity of regular tree languages
Given a Polish space X, recall that the class of Borel sets Borel(X) is obtained
from the open sets of X by the set-theoretic operations of complementation and
countable unions and that this class can then naturally be spread in a hierarchy of length ω1 , called the Borel hiearchy. But Borel sets are not the end of
the story. Next we have the projective sets, which are obtained from the Borel
sets by the operations of projection, or continuous image, and complementation. Indeed, although Borel(X) is closed by continuous pre-images, it is not
closed under continuous images. Analogously to the case of Borel sets, the class
Proj(X) of all projective sets of a Polish space X ramifies in a hierarchy of
length ω. More precisely, we have the following. If L ⊆ X × Y , the projection
of L is π ′′ (L) =: {x ∈ X : ∃y(x, y) ∈ L}. Note that π ′′ is continuous whenever
X and Y are topological spaces. We call a subset of a Polish space X analytic
if it is the projection of a Borel subset of X × N , where N is the Baire space
1 The
Borel hierarchy has already been introduced in Section 1.2.
5.1. TOPOLOGICAL HIERARCHIES
117
and X × N is equipped with the product topology. Thence, for every Polish
space X, the class of projective sets Proj(X) is obtained from the analytic sets
of X by the set-theoretic operations of complementation and projection. As for
the Borel sets, also this class can naturally be spread in a hierarchy of length ω,
called the projective hierarchy, where the class Σ11 (X) is constituted by all the
analytic subsets of X. By a famous theorem of Suslin, it is well-known that the
class of Borel subsets of a Polish space X is the class of all subsets of X which
are both analytic and co-analytic.
Let C be a class of sets in Polish spaces. A set L is C-hard if M ≤W L for
all M ∈ C, and C-complete if it C-hard and L ∈ C.
What we know about the topological complexity of regular tree languages?
First of all, they are all in the second projective ambiguous class ∆12 (cf. Chapter
10 of [104] for a proof of this fact), while if we look just at weakly recognizable
languages, they are all Borel and inhabit all the finite levels of the Borel hierarchy [94, 88, 8], and nothing more [51]. On the other hand, Niwinski and
Walukiewicz [101] provided a so called “Gap Theorem”, which implies that a
deterministic tree language is either on the level Π03 of the Borel hierarchy, or it
is Π11 -complete, and hence non Borel2 . The Gap Theorem actually gives an effective criterion for this dichotomy: a deterministic automaton recognizes a Π03
tree language iff its transition graph does not contain a certain forbidden pattern. Inspired by this result, Murlak [90] provided analogous forbidden patterns
for the remaining five Borel classes and thus proved decidability of the Borel
hierarchy for deterministic recognizable tree languages. The non-deterministic
case lacks decision procedures for the Borel and projective hierarchies. That is
to say we still do not have a general, effective, procedure for determining for
every regular tree language its position in the Borel and projective hierarchies.
As we will see in the next subsections, this is still the case if we restrict the class
of regular tree languages to those which are weakly recognizable.
5.1.2
The Wadge hierarchy of Borel sets of finite rank
The Wadge hierarchy consists of the collection of all sets of full binary trees
ordered by the Wadge reduction, and the Borel Wadge hierarchy is the restriction of the Wadge hierarchy to the Borel tree languages. In the sequel, since
we focus on weakly recognizable tree languages, we are particularly interested
in the Borel Wadge hierarchy.
In Chapter 1 we saw that the Wadge reduction ≤W is a pre-order. But can
we say something more about this order? First of all, as for parity games, it
can be asked whether when L ⊆ TΣ and M ⊆ TΣ′ are Borel sets, the Wadge
game GW (L, M ) is determined. Because a Wadge game is a special case of what
is called a Gale-Stewart game (cf. [70]), the positive answer comes from the
determinacy of Gale-Stewart games whose winning conditions are Borel, a very
famous result proved by Martin [85].
Theorem 5.1 (Borel Wadge Determinacy). L ⊆ TΣ and M ⊆ TΣ′ be two
Borel tree languages. Then GW (L, M ) is determined
2 Interestingly, since Büchi automata can recognize only Σ1 -sets of trees, this result provide
1
a very simple topological argument of the fact that these automata cannot recognize all regular
tree languages.
118
CHAPTER 5.
This result is the cornerstone of the description of the Borel Wadge hierarchy.
Indeed, the determinacy of Borel Wadge games induces the following corollaries:
the ≤W - antichains have length at most two, and the only incomparable tree
languages are - up to Wadge equivalence - of the form L and L∁ , for L nonself-dual. Since the Wadge reduction can be shown to be well-founded on Borel
sets, this means that - up to complementation and Wadge equivalence - the ≤W
provides a well-ordering on Borel sets. Formally, we have the following:
Proposition 5.2. L ⊆ TΣ and M ⊆ TΣ′ be two Borel tree languages. The
following properties hold.
(1) (Wadge’s Lemma) Either L ≤W M , or M ≤W L∁ .
(2) If L and M are incomparable, then L ≡W M ∁ .
(3) The ≤W -antichains have length at most two.
Proof.
(1) Either L ≤W M , or L 6≤W M . Suppose L 6≤W M . Then Duplicator
has no winning strategy in GW (L, M ). By determinacy, this means that
Spoiler has a winning strategy σ in this game. From this, we describe
a winning strategy for Duplicator in GW (M, L∁ ) as follows. On her first
move, regardless Spoiler’s move, Duplicator answers by σ applied to a skip
move. Then, Duplicator answers to every current position t[n] of Spoiler
by the move σ(t[n]). At the end of the play, the definition of σ ensures that
the full tree t played by Spoiler belong to M if and only if the tree played
by Duplicator by applying σ doesn’t belongs to L. Hence, Duplicator wins
the game GW (M, L∁ ), and M ≤W L∁ .
(2) If L 6≤W M and M 6≤W L, then the previous point implies that M ≤W L∁
and L ≤W M ∁ . Therefore, M ≤W L∁ and L∁ ≤W M , and finally L∁ ≡W
M.
(3) Let L, M and N be three tree languages such that L 6≤W M and M 6≤W
N . Then the first point implies that M ≤W L∁ and the second that
N ≤W M ∁ . Therefore, N ≤W M ∁ and M ∁ ≤W L, and by transitivity
N ≤W L.
Proposition 5.3 (Martin, Monk). The strict Wadge reduction <W is wellfounded on Borel tree languages.
Propositions 5.2 and 5.3 show that - up to complementation and Wadge
equivalence - the Borel Wadge hierarchy is a well-ordering. Therefore, there
exists a unique ordinal, called the height of the Borel Wadge hierarchy, and a
mapping dW from the Borel Wadge hierarchy onto its height, called the Wadge
degree, such that dW (L) < dW (M ) if and only if L <W M and dW (L) = dW (M )
if and only if L ≡W M or L ≡W M ∁ , for every pairs of Borel tree languages
L and M . The Borel Wadge hierarchy is thus a (huge) refinement of the Borel
hierarchy.
Recall that we have two kinds of tree languages: the self-dual and the non
self-dual ones. Another nice consequence of the Borel determinacy of Wadge
games is that every self-dual set can be described by translations of strictly ≤W smaller non self-dual sets (cf. [47, 48]). This means that we can concentrate
only on the non self-dual sets and recursively define the Wadge degree in such a
119
5.1. TOPOLOGICAL HIERARCHIES
way that sticks every self-dual set on a non self-dual set
in the hierarchy as follows:


1
dW (X) = sup {dW (Y ) + 1 | Y n.s.d. and Y <W X}


sup {dW (Y ) | Y n.s.d. and Y <W X}
lower at just one level
if X = ∅ or X = ∅c ,
if X is non-self-dual,
if X is self-dual.
Finally, one can show that the Borel Wadge hierarchy consists of an alternating succession of non self-dual and self-dual sets, as depicted in figure 5.1.
Figure 5.1: The Borel Wadge hierarchy: circles represent Wadge-equivalence
classes and arrows stand for the strict Wadge reduction between those. The
non self-dual sets and the self-dual ones located just one level above share the
same Wadge degree.
If we consider only the class ∆0ω of Borel sets of finite rank, the height of
the corresponding Wadge hierarchy is
ω1
··
ω·
sup ω1 1
n∈ω | {z }
n times
= ω1 ǫ 0 ,
the least fixpoint of the ordinal exponentiation of base ω1 (cf. [47, 48]).
Since we are interested in sets of trees which are recognizable (or equivalently
definable by a µ-formula), the question now is: can we described and possibly
decide the Borel Wadge hierarchy restricted to recognizable tree languages?
Thanks to a very nice work of Murlak [92], we know that the answer is positive
for deterministic tree languages. Unsurprisingly, the general problem is a very
difficult one and still remains widely open.
120
5.2
CHAPTER 5.
Weak alternating tree automata
In this part of the dissertation we will work with weak alternating automata
on full binary trees. Recall from Chapter 1 that a weak alternating parity tree
automaton is an alternating parity automaton, satisfying the condition that if
a state q is reachable from the state q ′ in the graph of the automaton, then the
parity associated to q ′ is less or equal to the parity associated to q. For a start,
we introduce some basic definitions that we also use in the next chapter. In the
second and third subsections we present and discuss the three corresponding
hierarchies: weak index hierarchy, Borel hierarchy and Wadge hierarchy.
5.2.1
Paths and loops
Given a weak alternating tree automaton A, a path in A is a sequence of states
and transitions
σn ,dn
σ1 ,d1
σ0 ,d0
q0 −−−→ q1 −−−→ q2 · · · qn −−−−→ qn+1
If there is such a path with q = q0 and q ′ = qn+1 , we say that q ′ is reachable
from q ′ . We say that a path is a loop if qn+1 = q0 . If there is a loop from a
state q, we say that this state is looping. If q is looping and Ω(q) is even (resp.
odd) we say that the loop in q is positive (resp. negative). Finally, we say that
σ0 ,d0
σn ,dn
a state p is replicated by q if there is a path q −−−→ q1 · · · qn −−−−→ p and a
σ0 ,d¯0
transition q −−−→ q. If q is reachable from the initial state qI , then we say that
Aq is sub-automaton of A, denoted by A > Aq . Clearly, the set {Aq : A > Aq }
is finite.
σn ,dn
σ1 ,d1
σ0 ,d0
Given a loop q −−−→ q1 −−−→ q2 · · · qn −−−−→ q, if it holds that qi = q, for
all 1 ≤ i ≤ n, we say that the loop is trivial. An automaton where all the loops
are trivial is called linear. An automaton without any loop is called strict.
Without loss of generality (except for the very special case of strict automata) we may assume that all states in the considered automata are productive save for one all-rejecting trivial-looping state ⊥ of odd priority, that every
state has an outgoing transition, and that all transitions are either productive
σ
or are of the form q −
→⊥, ⊥. Sometimes it will also be useful to have an all
accepting trivial-looping state ⊤ of even priority.
5.2.2
Weak index vs Borel rank
We have already seen that the index hierarchy of weak alternating tree automata, or weak index hierarchy, is strict. Concerning the topological hierarchies of weakly recognizable languages, we know since a long time that:
Proposition 5.4 ([88]). For every weak alternating automaton with index (0, n)
(resp. (1, n + 1)), it holds that L(A) ∈ Π0n (resp. L(A) ∈ Σ0n ).
We can immediately ask whether the other implication also holds, and therefore
whether for weakly recognizable tree languages the weak index hierarchy and
the Borel hierarchy coincide level by level. The answer is still unknown, but
in [93], it was conjectured that it should be positive, i.e., that:
(Weak index vs Borel rank conjecture) For every weakly recognizable tree
language L,
121
5.2. WEAK ALTERNATING TREE AUTOMATA
• L ∈ Σ0n iff there is a weak alternating automaton A of index (1, n+1)
such that L = L(A), and
• L ∈ Π0n iff there is a weak alternating automaton A of index (0, n)
such that L = L(A).
It was recently proved that the conjecture holds when restricted to languages
which are in addition deterministically recognizable [93]. But at our knowledge,
since then no other step towards a solution for Murlak’s conjecture has been
made.
Σ01 = [1, 2]
>
>
Σ02 = [1, 3]
>
>
∆01
∆02
= [0, 1] ∩ [1, 2]
Σ03 = [1, 4]· · ·
>
>
∆03
= [0, 2] ∩ [1, 3]
>
···
>
>
Π01 = [0, 2]
= [0, 3] ∩ [1, 4]
>
>
Π02 = [0, 1]
>
Π03 = [0, 3]· · ·
Figure 5.2: Representation of Murlak’s conjecture on the correspondence between weak index and Borel rank for weakly recognizable tree languages.
5.2.3
The Wadge hierarchy of weak alternating automata
What about the Borel Wadge hierarchy restricted to weakly recognizable languages? In [51], the authors provided a lower bound for its height. It is interesting and useful to explain a little bit more about this result. In order to do
so, let’s come back to the Borel Wadge hierarchy. Because of a remarkable correspondence between ordinal and set theoretical operations, Duparc has shown
in his PhD thesis [47] how to construct for any α strictly smaller than the least
fixpoint of the ordinal operation of exponentiation of base ω1 , a Borel set Ω(α)
of Wadge degree exactly α whose definition is isomorphic to the Cantor normal
form of base ω1 of this very same ordinal α. This means that we are able to
generate the Wadge hierarchy of Borel sets of finite rank from scratch, that is
starting from the empty set. This result can be important for describing the
Wadge hierarchies of some classes of word and tree automata. This is because,
on the one hand, in order to obtain lower bounds for the Wadge hierarchy of the
considered class of automata, it is often enough to check that some operations
are definable in the class, as is done by Duparc and Murlak [51] for weak alternating automata. On the other hand, it is sometimes possible to verify that a
certain pattern in the graph of the considered automaton corresponds exactly
to a certain set-theoretic operation. Because of the correspondence between
set-theoretic and ordinal operations, the Wadge degree of an automaton can
thence be obtained by induction on its structure. This strategy is for instance
the one successfully used by Murlak [92] in order to give an effective description
of the Wadge hierarchy of deterministic recognizable tree languages, and it is
122
CHAPTER 5.
also the one we use in the next chapter in the same aim but for the class of tree
languages recognized by, what we call, linear game automata.
Thus, let us have a look at some of these set-theoretic operations3 .
We start defining four basic operations on sets of trees, that will also be
useful in the next chapter. Let L, M ⊆ TΣ , and assume that Σ contains at
least two letters, a and b. Define alternative (∨), disjunctive product (⋄), and
conjunctive product ( 2 ) as
L ∨ M = {t : t(ε) = a , t.0 ∈ L or t(ε) 6= a , t.0 ∈ M } ,
L ⋄ M = {t : t.0 ∈ L or t.1 ∈ M } ,
L 2 M = {t : t.0 ∈ L and t.1 ∈ M } .
Multifold alternatives and parallel compositions are performed from left to right,
e.g., L1 ∨ L2 ∨ L3 ∨ L4 = (((L1 ∨ L2 ) ∨ L3 ) ∨ A4 ). It is easy to see that these three
operations define associative and commutative operations on Wadge equivalence
classes.
To allow easier reading, we sometimes write Lhki for the set L
· · ⋄ L}, and
| ⋄ ·{z
k times
analogously L[k] for |L 2 .{z
. . 2 L}.
k times
Another useful operation on sets if the following. Let L, M ⊆ TΣ . We define
the set L → M as the set of trees t ∈ TΣ∪{a} , with a ∈
/ Σ, satisfying any of the
following conditions:
• if a = t(11n ) for all n, then t.0 ∈ L,
• if 11n is the first node on the path 11∗ not labeled by a, then t.11n 0 ∈ M .
A player in charge of L → M is like a player in charge of L endowed with an
extra move, which can be used only once, that erases everything played before.
Then she can restart the play being in charge of M . We say that a non-self dual
set L ⊆ TΣ is initializable when L ≥W L → L.
Sum and supremum : Suppose that L, M ⊆ TΣ . We define the set M + L
as L → M ∨ M ∁ . From the point of view of the player in charge of the set
M + L in a Wadge Game, everything goes as if she was starting the game being
in charge of L. So, provided she plays in such a way that a always holds in the
rightmost branch of the tree, the question whether the resulting infinite tree she
will have produced at the end of the run belongs to M + L or not reduces to the
question whether the tree starting from the left son of the root belongs to L or
not. But at any moment of the run she can play a node 11n not labelled with
a. Then, everything looks like the whole (finite) tree played since the beginning
of the game is erased and he is now in charge of: M if a is the label of the node
(11n 1), M ∁ else.
The following remark ensures that the set-theoretic operation + is the exact
counterpart of the ordinal sum on Wadge degrees of Borel sets.
Remark 5.5 ([49, 47]). Let L, M ⊆ TΣ be two non self-dual Borel sets of full
binary trees. Then dW (L + M ) = dW (L) + dW (M ).
3 Duparc originally defined these operations for sets of infinite words. The operations on
tree languages we present here are their straightforward translation for trees given in [51].
5.2. WEAK ALTERNATING TREE AUTOMATA
123
As for alternative, it is easy to see that sum defines associative and commutative
operations on Wadge equivalence classes.
The next operation is a generalization of ∨ and +. Let λ < ω1 , and Lκ ⊆
TΣ∪{b} for any κ < λ. Fix any 1 − 1 map f : ω → λ. Thus, define sup−
κ<λ Lκ as
the set of trees t ∈ TΣ∪{b} satisfying the following conditions for some k:
• 0k is the first node on 1∗ labeled with b,
• t.0k 1 ∈ Lf (k) .
Intuitively, a player in charge of sup−
κ<λ Lκ is given the choice between the Lκ ’s.
The decision is determined by the number of labels different from b played on
the leftmost branch of the tree before the first b. If the player keeps not playing
b forever on the leftmost branch, the tree will be rejected.
−
n
Define also sup+
κ<λ Lκ as supκ<λ Lκ ∪ {t : ∀n t(1 ) 6= b}. The difference
from the previous operation is that now, when the player does not plays b on
the leftmost branch, the obtained tree is accepted. Note that the operations are
dual:
∁
+
−
sup Lκ
= sup L∁κ
κ<λ
κ<λ
The following fact ensures that, on Borel sets, the set-theoretic sup is the counterpart of the ordinal supremum on Wadge degrees.
Remark 5.6 ([49, 47]). Let (Lκ )κ<λ and (Mκ )κ<λ be two countable families of
non self-dual Borel sets of full binary trees. Then
+
−
κ<λ
κ<λ
dW ( sup Lκ ) = dW ( sup Lκ ) = sup dW (Lκ ).
κ<λ
Countable multiplication : With the help of ordinal sum and countable
supremum we easily define the set-theoretic counterpart of the countable multiplication as an iterated sum. Let L ⊆ TΣ . Inductively we define:
• L • 1 = L,
• L • (α + 1) = (L • α) + L,
• L • λ = sup+
κ∈λ L • κ when λ is some limit countable ordinal.
As for the previous operations, we remark that the set-theoretic countable
multiplication is the exact counterpart of the ordinal operation of countable
product on Wadge degrees.
Remark 5.7 ([49, 47]). Let L ⊆ TΣ be a non self dual Borel set. Then for every
countable ordinal λ, dW (L • λ) = dW (L) λ
From the player’s point of view when involved in Wadge Games, being in
charge of a set of the form L • λ is like a player being in charge of L with
the additional option to restart the run at any moment being in charge of its
complement L∁ instead of L and start again and again replacing alternatively L∁
by L and L by L∁ , provided that at every such changing the player decreases the
ordinal λ. Therefore, during the run, this procedure will produce a decreasing
finite sequence of ordinals, preventing her from initializing the game indefinitely.
124
CHAPTER 5.
The last operation, called the action of a closed set, is much more subtle
and involved than the previous ones and will not explained here (cf. [49, 47, 51]).
What is important is that this operation on Borel sets was shown to correspond
(almost exactly) to the exponentiation of base ω1 : α 7→ ω1α .
By showing that the operations of sum, multiplication by ω and the action
of a closed set are all definable by a weak alternating tree automaton, Duparc
and Murlak [51] were thus able to infer that the Wadge hierarchy of weakly
recognizable tree languages has height at least ǫ0 . It is still unknown whether
the bound is tight, not to mention the problem of deciding the Wadge degree of
a weakly recognizable language.
5.3
Summarizing remarks
In the next chapter we will study the index, Borel and Borel Wadge hierarchies
of a subclass of weak alternating automata capturing a very weak form of alternation. In this aim, in this introductory chapter we briefly discussed those
three hierarchies in the case of weakly recognizable tree languages.
We already know that the weak index hierarchy is strict and that weak alternating automata can recognize languages arbitrary high in the finite Borel
hierarchy. In particular, it is known that a weak automaton of index (0, n),
resp. (1, n + 1), is in the Borel class Π0n , resp. in Σ0n . It is thus immediate to
ask whether the other implication also holds, and therefore whether for weakly
recognizable tree languages the weak index and the Borel rank coincide. The answer is still unknown, but Murlak [93], after verifying that this correspondence
holds when restricted to languages which are in addition deterministically recognizable, conjecture that it should be positive. In the next chapter we show
that the conjecture is true for the considered subclass of weak alternating automata, which is orthogonal to the the class of deterministic tree automata but
recognize languages of arbitrary high finite Borel rank. Moreover we verify that
both hierarchies, weak index and Borel, are decidable.
In Subsection 5.1.2, we introduced the Borel Wadge hierarchy. Thanks to
the work of Wadge [125] and Duparc [47, 49], we know that this hierarchy can
be generated from the empty set by way of a finite number of set-theoretic
operations, which correspond exactly to some ordinal operations (Subsection
5.2.3). This means that for any α strictly smaller than the height of the Borel
Wadge hierarchy, we can construct a Borel set Ω(α) of Wadge degree exactly
α whose definition is isomorphic to the Cantor normal form of base ω1 of this
very same ordinal α. This observation can be important for the study of the
hierarchy when restricted to tree languages recognized by a certain class of
automata. For example, by showing that some operations are definable by weak
alternating tree automata, Duparc and Murlak [51] were able to infer that the
Wadge hierarchy of weakly recognizable tree languages has height at least ǫ0 .
In the next chapter, some properties of the correspondence between operations
on sets and operations on ordinals are exploited in order to give an effective
characterization of the Wadge hierarchy of the considered class of automata.
Chapter 6
Decidable Hierarchies for
Linear Game Automata
This chapter is based on a joint work with Jacques Duparc and Filip Murlak
[50].
6.1
Preliminary remarks
Alternating tree automata (or equivalently the modal µ-calculus), are notorious
for the lack of decision procedures for classical hierarchies like the MostowskiRabin hierarchy (resp. the fixpoint alternation hierarchy), the Borel hierarchy,
or the Wadge hierarchy. The reason for this is that when we move from infinite
words to infinite trees, deterministic and non-deterministic modes of computation highly diverge. Topologically this is shown by the fact that all recognizable
tree languages are in ∆12 , while deterministic recognizable tree languages are
either Π11 -complete or in the third level of the Borel hierarchy, as shown by
Niwinski and Walukiewicz [101].
In this chapter we propose a novel class of automata capturing an interesting
aspect of alternation and prove that all corresponding hierarchies mentioned
above are decidable. The class we advocate, linear game automata, LGA for
short, is obtained by taking linear automata and restricting the alternation to
the choice of a path in the input tree. Even if linear game automata are weaker
than linear automata, they preserve most alternations related to the branching
structure. Evidence for the expressivity of LGA comes from their topological
properties: they recognize sets of arbitrarily high finite Borel rank, and their
Wadge hierarchy has the height (ω ω )ω .
As we have already pointed out, these automata are very weak; so this is
just a very first step on the way to the complete understanding of alternating
automata, or the modal µ-calculus, on infinite trees. This notwithstanding,
computing the Wadge degree of a LGA is much more involved than calculating the Wadge degree of an ω-word automaton and even a deterministic tree
automaton. If the shape of these automata seems to bring down the calculation to the decomposition of nested chains, alternation (even if very weak)
makes everything much harder to compute, more expressive and complicated.
We also believe that the notion of game automata is well suited to take us fur125
126
CHAPTER 6.
ther. Indeed, the next step is to consider weak game automata, then strong
game automata, which inhabit every level of the (strong) index hierarchy and
subsume deterministic languages.
The structure of the chapter is the following. In the next section we formally
introduce the class of linear game automata. After providing a normal form for
those automata, in Section 6.3 we show the the problem of determining the
Borel rank of a LGA-recognizable language is decidable. In Section 6.4, the
same problem but for the weak index is solved by an exact reduction to the
Borel rank problem. Then, before the concluding remarks, we give a complete
effective description of the corresponding Wadge hierarchy.
6.2
Linear game automata
A linear game automaton (LGA) is a linear alternating automaton where the
transition relation is a (total) function δ : Q × Σ × {0, 1} → Q.
σ
In the remaining of the chapter, we often write q −
→ q0 , q1 if δ(q, σ, 0) = q0
and δ(q, σ, 1) = q1 . Recall that without loss of generality, we assume that:
• there is no trivial state, i.e., if q ∈ Q is such that Aq ≡ ⊤ (resp. Aq ≡ ⊥),
then q = ⊤ (resp. q = ⊥),
σ
• there is no trivial transition, i.e., if p ∈ Q∀ , and p −
→ q, ⊥, then q = ⊥
(dually for p ∈ Q∃ ).
Moreover, by convention, ⊤ is a looping state of even rank, and ⊥ is a looping
state of odd rank. Recall also that if q is looping and the priority Ω(q) is even
(resp. odd) we say that q is a positive (resp. negative) looping state.
LGA are closed under complementation. The usual complementation procedure, that increases the ranks by one and swaps existential and universal states
turns LGA into LGA. However, LGA are not closed under union nor intersection.
Given σ ∈ Σ, the language Lσ = {t ∈ TΣ : t(0) = t(1) = σ} is LGA-recognizable,
but Lσ ∪ Lσ′ is not.
Linear game automata clearly do not recognize all deterministic tree languages. However, not every tree language recognized by an LGA automaton
is deterministic recognizable. An argument for this fact comes from the topological complexity of LGA-recognizable languages. Indeed, while from the Gap
Theorem [101], a deterministic tree language is either on the level Π03 of the
Borel hierarchy or it is not Borel, linear game automata can recognize tree languages of any finite Borel rank. This is because, as will be noticed in Section
6.3, some canonical LGA-recognizable languages coincide with the sets used by
Skurczynski to prove the existence of weakly recognizable languages of each
finite Borel rank.
6.2.1
A normal form
We now provide a useful normal form of LGA. Let L, M be tree languages over
Σ containing at least two letters, a and b. The family of languages recognized
by LGAs is closed under the operations of alternative (∨), disjunctive product
(⋄), and conjunctive product ( 2 ) introduced in Chapter 5. In particular, the
operations have natural counterparts on automata. We write A ∨ B to denote
127
6.2. LINEAR GAME AUTOMATA
the automaton recognizing L(A) ∨ L(B), and similarly for ⋄ and 2 . Multifold
alternatives are performed from left to right, e.g., A1 ∨ A2 ∨ A3 ∨ A4 = (((A1 ∨
A2 )∨A3 )∨A4 ). It is easy to see that these three operations define associative and
commutative operations on Wadge equivalence classes. To allow easier reading,
we sometimes write Ahki for the automaton A
· · ⋄ A}, and analogously A[k]
| ⋄ ·{z
k times
for |A 2 .{z
. . 2 A}.
k times
Lemma 6.1. Each LGA is Wadge equivalent to an LGA over the alphabet {a, b}.
Proof. We proceed by induction on the number of states. Let C be an LGA. If C
has only one state, the claim follows trivially. Suppose C has several states. We
may assume w.l.o.g. that its initial state of C, q0 , is existential. Suppose that
bj
a
c
i
k
q0 , q0
pi , p′i , q0 −→ q0 , rj and q0 −→
the transitions of C starting in q0 are q0 −→
with Σ = {a1 , . . . , aℓ ; b1 , . . . , bm ; c1 , . . . , cn }. Then C is Wadge equivalent to
bj ,0
ck ,∗
⊥
, hq0 i r
/ Cr1 ∨ · · · ∨ Crm
RRR
RRR bj ,1
~
~
RRR
~~
RRR
ai ,1
~~ ai ,0
RR(
~
~
(Cp1 ⋄ Cp′ ) ∨ · · · ∨ (Cpℓ ⋄ Cp′ )
1
ℓ
By induction hypothesis, there exist automata Ai , A′i , Bj over {a, b}, such
that Ai ≡W Cpi , A′i ≡W Cp′i , and Bj ≡W Crj . Let A = (A1 ⋄A′1 )∨· · ·∨(Aℓ ⋄A′ℓ )
and B = B1 ∨ · · · ∨ Bm . Further, we see that if A ∨ B ≡W ⊤, then C is Wadge
equivalent to the automaton on the left below and otherwise to the one on the
right:
⊤
b,0
b,0
, hq0 i
/
BBb,1 ⊥
|
B
|
BB
||
BB
|| a,0
a,1
B!
|
~|
, hq0 i
/
GG b,1 B ∨ A
|
G
|
GG
||
G
|| a,0
a,1 GG
|
G#
~|
⊤
⊥
A
From now on we work with automata over {a, b}, unless explicitly stated otherwise.
A looping state q of a linear game automaton A is
• restrained if it is an existential positive state or a universal negative state,
• unrestrained if it is an existential negative state or a universal positive
state.
Examining the proof of Lemma 6.1, we see that in fact, each nontrivial looping
state falls into exactly one of the categories shown below (+ means even rank,
− means odd rank).
128
CHAPTER 6.
Restrained
B0
Unrestrained
a,0
a,0
, h+i a,1 / A
{ CCC
CC
{{
{
C
{{ b,0
b,1 CC
{
}{
!
, h−i a,1 / A
{ CCC
CC
{{
{
CC
{{ b,0
b,1
C!
{
}{
B1
B0
a,0
B0
B1
a,0
- [−] a,1 / A
BB
|
BB
|
BB
||
|
BB
|
b,0
b,1
|
~|
B1
B0
- [+] a,1 / A
BB
|
BB
|
BB
||
|
BB
|
b,0
b,1
|
~|
B1
The term “restrained” is associate to existential positive looping states and
universal negative looping states because, as we will see, the associated gadget
acts like a sort of automata counterpart of the set-theoretic operation of sum. On
the contrary, the term “unrestrained” is associate to existential negative looping
states and universal positive looping states because the associated gadget can
act like the automata counterpart of the more powerful set-theoretic operation
of exponentiation of base ω1 .
A node q of each of the above kinds may be seen as an action over triples of
LGAs; we denote by q(A, B0 , B1 ) the automaton being the result of the action q
on A, B0 , B1 , e.g., [+](A, B0 , B1 ) or h−i(A, B0 , B1 ). Often we use a shorthand
[ǫ](A, B) = [ǫ](A, B, ⊤), hǫi(A, B) = hǫi(A, B, ⊥) for ǫ = + or ǫ = −.
6.3
Deciding the Borel hierarchy
In this section we show that the problem of determining the Borel rank of a tree
language recognized by a linear game automaton is decidable. We approach this
issue by using the technique of difficult patterns in the graph of an automaton.
The general idea is the following. For every Borel class of finite rank we identify
a certain pattern satisfying the following condition: if the pattern appears in an
automaton A, then it provides a reduction of some difficult language to L(A);
otherwise L(A) is in the considered class. This technique was used by Wagner
[126] in his solution of the general problem of continuous reductions between
ω-languages, and it has been successfully extended to the case of deterministic
recognizable tree languages by Niwinski and Walukiewicz [101, 102], and Murlak
[90].
6.3.1
Patterns menagerie
The basis for the procedure computing the Borel rank of a given LGA-recognizable
language is a characterization in terms of difficult patterns. We define (0, n)pattern, and (1, n + 1)-pattern by induction on n:
• a (0, 1)-pattern is a negative looping state reachable from a positive looping
state,
• a (1, 2)-pattern is a positive looping state reachable from a negative looping
state,
129
6.3. DECIDING THE BOREL HIERARCHY
• a (0, n + 1)-pattern is a (1, n + 1)-pattern replicated by a universal positive
state,
• a (1, n + 2)-pattern is a (0, n)-pattern replicated by an existential negative
state.
We also define canonical automata, KnΣ and KnΠ , corresponding to the patterns:
K1Π = [+](⊤, ⊥, ⊥),
Π
Kn+1
= [+](KnΣ , ⊥, ⊥),
K1Σ = h−i(⊥, ⊤, ⊤),
Σ
Kn+1
= h−i(KnΠ , ⊤, ⊤).
As an example, the next figure shows the structure of the canonical automaton
K4Σ (instead of the signs “+” and “-”, we write the parity associated to the
states, and instead of ⊥ and ⊤, we sketch explicitly the all-accepting and allrejecting looping states):
a,0
/
a,0
h1i
a,0
/
a,1
[2]
b
a,1
h3i
b
/
a,1
[4]
C [3]
[4]
C [4]
∗
a,1
/
b
∗
∗
b
C [2]
∗
/
a,0
C [5]
∗
The tree languages recognized by the above canonical automata coincide
with the sets used by Skurczynski to prove the existence of weakly recognizable
languages of each finite Borel rank.
Proposition 6.2 ([114]). For every n > 0,
L(KnΣ ) is Σ0n -complete and L(KnΠ ) is Π0n -complete.
Skurczynski’s result follows by straightforward induction from the following
easy lemma. For v ∈ {0, 1}∗ and U ⊆ TΣ , let vU = {t ∈ TΣ : t.v ∈ U }.
Lemma 6.3. For each n > 0
(1) if Ui is Σ0n -hard for i < ω,
(2) if Vi is Π0n -hard for i < ω,
6.3.2
T
i∈ω
0i 1Ui is Π0n+1 -hard;
i∈ω
0i 1Vi is Σ0n+1 -hard.
S
Effective characterization
Since any Borel class is closed under finite unions and finite intersections, we
have:
Proposition 6.4. Let K be a complete set for some Borel class of finite rank.
For every k, if Ui ≤W K for 0 ≤ i ≤ k, then
S
T
(1) ki=0 0i 1Ui ≤W K,
(2) ki=0 0i 1Ui ≤W K,
and if Vi <W K for 0 ≤ i ≤ k, then
S
T
(3) ki=0 0i 1Vi <W K,
(4) ki=0 0i 1Vi <W K.
130
CHAPTER 6.
Analogously, since Σ0n is closed under countable unions, and Π0n is closed under
countable intersections, we obtain the following result.
Proposition 6.5.
(1) Let KS be a Σ0n -complete set. If for every i ∈ ω it holds that Ui ≤W K,
then i∈ω 0i 1Ui ≤W K.
(2) Let KT be a Π0n -complete set. If for every i ∈ ω it holds that Ui ≤W K,
then i∈ω 0i 1Ui ≤W K.
We now apply these properties to characterize the topological power of looping nodes in an LGA.
Lemma 6.6. Let A, B0 , B1 , C be LGA such that C = q(A, B0 , B1 ). Then
(1) if q is a negative looping node and L(A), L(Bi ) ≤W L(K1Σ ), then L(C) ≤W
L(K1Σ );
(2) if q is a positive looping node and L(A), L(Bi ) ≤W L(K1Π ), then L(C) ≤W
L(K1Π ).
Proof. We prove the first claim, the second follows by duality. Assume that q is
a negative looping node and L(A), L(Bi ) ≤W L(K1Σ ). Let us describe a winning
strategy for Duplicator in GW (L(C), L(K1Σ )).
Suppose that q = h−i. As long as Spoiler plays a on the leftmost branch,
from the point of view of a player in a Wadge game, he is like a player being in
charge of a countable union of open sets, and therefore Duplicator have just to
apply the winning strategy given by Proposition 6.5 (1). If Spoiler finally plays
a b in the kth round, he is like a player in charge of a finite union of open sets.
Thus Duplicator switches to playing rejecting in every subtree rooted in 0i 1 for
i < k, and in the subtree rooted in 0k applies the winning strategy given by
Proposition 6.4 (1). Hence, L(C) ≤W L(K1Σ ).
Suppose that q = [−]. As long as Spoiler plays a on the leftmost branch,
Duplicator plays rejecting in every subtree rooted in 0i 1. If Spoiler finally plays
a b in the kth round, from the point of view of a player in a Wadge game, he is
like a player in charge of a finite intersection of open sets. Thus in the subtree
rooted in 0k Duplicator applies the winning strategy given by Proposition 6.4
(2). Hence, in this case too L(C) ≤W L(K1Σ ).
Lemma 6.7. Let A, B0 , B1 , C be LGA such that C = q(A, B0 , B1 ), and q is a
restrained looping node. For n ≥ 2
(1) if L(A), L(Bi ) <W L(KnΣ ), then L(C) <W L(KnΣ );
(2) if L(A), L(Bi ) <W L(KnΠ ), then L(C) <W L(KnΠ ).
Proof. As for the previous lemma, it is enough to prove the first claim, the
second follows by duality. Suppose that q = h+i. We describe a winning
strategy for Duplicator in GW (L(C), L(KnΣ )). If Spoiler plays a on the leftmost
branch, Duplicator plays accepting in the subtrees rooted in nodes 0i 1. If Spoiler
finally plays a b in the kth round, Duplicator switches to playing rejecting in
every subtree rooted in 0i 1 for i < k, and in the subtree rooted in 0k applies
the winning strategy given by Proposition 6.4 (1). Hence, L(C) ≤W L(KnΣ ).
6.3. DECIDING THE BOREL HIERARCHY
131
To obtain strictness of the inequality, we describe a winning strategy for
Spoiler in GW (L(KnΣ ), L(C)). As long as Duplicator skips or plays a on the
leftmost branch, Spoiler plays rejecting in the subtrees rooted in 0i 1. If in the
kth round Duplicator finally plays b on the leftmost branch, Spoiler continues
playing rejecting in every subtree rooted in 0i 1 for i ≤ n, and in the subtree
rooted in 0n+1 applies the winning strategy given by Proposition 6.4 (3).
For q = [−] the proof is analogous, only uses Proposition 6.4 (2) and (4).
Lemma 6.8. Let A, B0 , B1 , C be LGA such that C = q(A, B0 , B1 ), and q is an
unrestrained looping node. Let n ≥ 2. If q = h−i, then
Σ
1. if L(A) ≤W L(Kn−1
), and L(Bi ) <W L(KnΣ ), then L(C) <W L(KnΣ );
Σ
Σ
2. if L(A), L(Bi ) ≤W L(Kn−1
), then L(C) ≤W L(Kn−1
);
Π
3. if L(A) ≥W L(Kn−1
), then L(C) ≥W L(KnΣ );
and if q = [+], then
Π
4. if L(A) ≤W L(Kn−1
), and L(Bi ) <W L(KnΠ ), then L(C) <W L(KnΠ );
Π
Π
5. if L(A), L(Bi ) ≤W L(Kn−1
), then L(C) ≤W L(Kn−1
);
Σ
6. if L(A) ≥W L(Kn−1
), then L(C) ≥W L(KnΠ ).
Proof. Use an argument similar to the proof of Lemma 6.7 to infer (1) and (4)
from Proposition 6.5, and (3) and (6) from Lemma 6.3, and use an argument
similar to the proof of Lemma 6.6 to infer (2) and (5) from Propositions 6.4 and
6.5.
The main theorem of this part follows from Lemmas 6.6, 6.7 and 6.8 by
induction on the structure of the automaton. And as a corollary, we obtain the
first decidability result.
Theorem 6.9. For every n ≥ 1 and every linear game automaton C
(1) L(C) is Π0n -hard iff C contains a (0, n)-pattern.
(2) L(C) is Σ0n -hard iff C contains a (1, n + 1)-pattern;
Proof. Recall that we always assume that all LGA have no trivial states and no
trivial transitions. Moreover we assume that C is over the alphabet {a, b} and
it is in the normal form given by the proof of Lemma 6.1. The reasoning for
the general case is analogous. We proceed by induction on the structure of the
automaton C. If C has only one state, both claims follow trivially. Suppose that
C has more than one state. Then, observe that if C does not contain neither a
(0, 1)-pattern nor (1, 2)-pattern, L(C) ∈ ∆01 . Hence, we may assume that C has
either a (0, 1) or (1, 2)-pattern. If the initial state of C is not looping, the claims
follow easily by applying the induction hypothesis. Let C = q(A, B0 , B1 ). We
first verify Point (1).
⇒:
Suppose that L(C) is Π0n -hard. Consider first the case when n = 1. We
reason towards a contradiction. So let assume that C does not contain
any (0, 1)-pattern but that it contains a (1, 2)-pattern, meaning that the
parity of q is odd. By induction hypothesis, L(A), L(B1 ) and L(B2 ) can
132
CHAPTER 6.
at most be Σ01 -complete. But by Lemma 6.6 (1), this implies that L(C)
cannot be Π01 -hard, a contradiction. Therefore C contains a (0, 1)-pattern.
Assume now that n > 1. If q is a restrained looping node, then by Lemma
6.7 (2), either A or B1 or B2 is Π0n -hard. By induction hypothesis C
contains a (0, n)-pattern. If q is an unrestrained node we have two cases
to consider: (i) q is an universal positive node, (ii) q is an existential
Π
negative node. In case (i), by Lemma 6.8 (4) either L(A) >W L(Kn−1
) or
0
0
L(Bi ) is Πn -hard, with i ∈ {0, 1}. If either L(B0 ) or L(B1 ) is Πn -hard,
by applying the induction hypothesis we have that C contains a (0, n)pattern. Suppose that neither L(B0 ) nor L(B1 ) is Π0n -hard, but L(A) >W
Π
Π
L(Kn−1
). Because by Proposition 6.2 L(Kn−1
) is Π0n−1 -complete, this
0
means that L(A) is Σn−1 -hard. By induction hypothesis A contains a
(1, n)-pattern, and therefore C contains a (0, n)-pattern. For case (ii) we
reason as follows. If either A or B0 or B1 contains a (0, n)-pattern, then C
contains a (0, n)-pattern too. On the contrary, assume that neither A nor
B0 nor B1 contain a (0, n)-pattern. By induction hypothesis neither L(A)
nor L(B0 ) nor L(B1 ) are Π0n -hard. But this implies that L(A), L(B0 )
and L(B1 ) can at most be Σ0n -complete. Since by assumption L(C) is
Π0n -hard, by Lemma 6.8 (2) we obtain a contradiction. Therefore either
A or B0 or B1 must contain a (0, n)-pattern.
⇐:
For the other direction, assume that C contains a (0, n)-pattern. Suppose
first that either A or B0 or B1 contains a (0, n)-pattern. Then by induction
hypothesis, either L(A) or L(B0 ) or L(B1 ) is Π0n -hard, and so is L(C).
Suppose that neither A nor B0 nor B1 contain a (0, n)-pattern. Assume
that n = 1. Given the assumptions, q must be a positive looping node,
otherwise either A or B0 or B1 would contain a (0, 1)-pattern. Thence,
one of the following possibilities holds:
• either: q is a positive universal node, L(B0 2 B1 ) ≥W ∅ and L(A) ≥W
∅∁ ,
• or: q is a positive universal node, B0 2 B1 = ∅∁ and L(A) ≥W ∅ ∨ ∅∁ ,
• or: q is a positive existential node, and L(B0 ⋄ B1 ), L(A) ≥W ∅.
In each case clearly L(C) ≥W L(K1Π ) and therefore by Proposition 6.2
L(C) is Π01 -hard. Assume now that n > 1. If q is a restrained looping
node or an unrestrained existential negative looping node, then either A
or B0 or B1 must contain a (0, n)-pattern, a contradiction. Thus, q is
an unrestrained universal positive looping node. This means that either
B0 or B1 contains a (0, n)-pattern, or A contains a (1, n)-pattern. In
the first case, by induction hypothesis either L(B0 ) or L(B1 ) is Π0n -hard,
and therefore L(C) is also Π0n -hard. In the second case, by induction
hypothesis L(A) is Σ0n−1 -hard, and by applying Lemma 6.8 (6) we obtain
that L(C) is Π0n -hard.
The proof for Point (2) is analogous.
Corollary 6.10. The problem of calculating the exact position in the Borel
hierarchy of a language recognized by a linear game tree automaton is decidable
(in polynomial time if the productive states are given).
6.4. WEAK INDEX OF LGA-RECOGNIZABLE SETS
6.4
133
Weak index of LGA-recognizable sets
In [93] it was conjectured that for weakly recognizable tree languages the weak
index hierarchy and the Borel hierarchy coincide, i.e., that a weakly recognizable
tree language is in Σ0n (resp. Π0n ) iff it can be recognized by a weak alternating
automaton of index (1, n + 1) (resp. (0, n)). It has long been known that one
implication holds [88], that is if a weak alternating automaton A have index
(0, n) (resp. (1, n + 1)), then L(A) ∈ Π0n (resp. L(A) ∈ Σ0n ). It was also proved
recently that the conjecture holds when restricted to languages which are in
addition deterministically recognizable [93]. We refine this result by showing
that the conjecture also holds for languages recognizable by LGA.
Theorem 6.11. For languages recognizable by LGA, the Borel hierarchy and the
weak index hierarchy coincide.
Proof. For simplicity we assume that all automata are in the normal form given
by the proof of Lemma 6.1. Extending the proof to the general case is easy.
By duality it is enough to consider Π0n classes. By Proposition 5.4 it is
suffices to show that for each LGA C with L(C) ∈ Π0n there exists an equivalent
weak alternating automaton of index (0, n). We proceed by induction on the
structure of the automaton.
The case n = 0 is trivial. Suppose that n = 1. By Theorem 6.9, C does not
contain an accepting loop reachable from a rejecting loop. It is enough to set
the rank of all states reachable from odd looping states to 1 and the rank of the
remaining states to 0 to obtain an equivalent automaton of index (0, 1).
Suppose that n ≥ 2. If the initial state of C is not looping, the claim follows
easily from the induction hypothesis. Suppose that q0 is a looping node, and C
is of the form
a,0
B0
+ (i) a,1 / A
AA
}
AA b,1
b,0 }}
AA
}
}
AA
}
~}}
B1
We can treat C as a weak alternating automaton and transform it into an
equivalent one of index (0, n). Clearly, it must hold that L(A), L(B) ∈ Π0n and
by induction hypothesis we may assume that A, B0 , B1 have index (0, n). If
i = 0, the claim follows trivially. For (i) = [1], the equivalent weak alternating
automaton of index (0, n) is shown below.
a,0
[0]
AA
AAε
~~
~
AA
~
~
A ~
~ a,1
~
, [1]
/⊤
[0]
ε
b,∗
b,0
⊤
a,0
/
BBa,1 A
BB
BB
BB
b,1
B0
B1
To prove the equivalence, observe that the left-hand component checks that
finally b occurs on the leftmost branch, and the right-hand component checks
134
CHAPTER 6.
the condition A until the first b occurs, and after that checks the conditions B0
and B1 .
Finally, suppose that (i) = h1i. By Theorem 6.9, C contains (1, n + 1)pattern, which implies that A contains no (0, n − 1)-pattern. By induction
hypothesis we may assume that A has index (1, n). Recall that B0 and
B1 have index (0, n). The corresponding equivalent weak alternating automaton is shown below.
a,0
[0]
AA
AA ε
}}
}
AA
}
}
AA
}
~}}a,1
+ h1i
/A
h0i
ε
b,∗
⊤
a,0
/
BBa,1 A
BB
BB
b,0
BB
b,1
!
B0
B1
The left-hand component takes care of the situation, when b never occurs
on the leftmost path. If b does occur, this component is trivially accepting, but
the right-hand component provides the appropriate semantics.
Combining Theorem 6.9 and Theorem 6.11 we obtain the second decidability
result.
Corollary 6.12. The problem of calculating the exact position in the weak
index hierarchy of a language recognized by a LGA is decidable (in polynomial
time if the productive states are given).
6.5
The Wadge hierarchy of linear game automata
In this section we provide a complete description of the Borel Wadge hierarchy
restricted to LGA recognizable tree languages. In particular we show that the
problem of determining the Wadge degree of a tree language recognized by a
linear game automaton is decidable. In this aim, we first introduce the classical
finite Hausdorff-Kuratowski, or difference, hierarchy and state some of its basic
properties. This hierarchy is important because the whole Wadge hierarchy of
linear game automata is, in some sense, built upon it. Then, in Subsection
6.5.2 we provide a family of canonical LGA recognizable tree languages. In
Subsection 6.5.3 we verify that those languages are complete for every level of a
certain normal form, thus obtaining a lower bound for the corresponding Wadge
hierarchy. That the bound is tight is also proved in Subsection 6.5.3. This is
done by showing that the previous degrees are closed under all the ordinal
operations corresponding to an action of a node. By relying on this fact, we
finally verify that the Wadge degree of any LGA-recognizable tree language can
be computed.
6.5.1
The difference hierarchy
For a Borel class Σ0n , the finite Hausdorff-Kuratowski, or difference, hierarchy
is defined as D1 (Σ0n ) = Σ0n and Dk (Σ0n ) = {U \ V : U ∈ Σ0n , V ∈ Dk−1 (Σ0n )}.
6.5. THE WADGE HIERARCHY OF LINEAR GAME AUTOMATA
135
Let Dk (Σ0n ) denote the dual class. Recall that this is not the same as Dk (Π0n ).
Indeed, D2k+1 (Π0n ) = D2k+1 (Σ0n ) and D2k (Π0n ) = D2k (Σ0n ). We have
D2k (Σ0n ) = {U1 ∩ V1∁ ∪ · · · ∪ Uk ∩ Vk∁ } ,
D2k+1 (Σ0n ) = {U1 ∩ V1∁ ∪ · · · ∪ Uk ∩ Vk∁ ∪ U } ,
∁
D2k (Σ0n ) = {U1 ∩ V1∁ ∪ · · · ∪ Uk−1 ∩ Vk−1
∪ U ∪ V ∁} ,
D2k+1 (Σ0n ) = {U1 ∩ V1∁ ∪ · · · ∪ Uk ∩ Vk∁ ∪ V ∁ } ,
where the sets U, V, Ui , Vi range over Σ0n . From this characterization one easily
obtains the following table of the operation ⋄. For n > 0 let Sn (k) be a Dk (Σ0n )complete set, and let Pn (k) be a Dk (Σ0n )-complete set.
Lemma 6.13. For each n > 0, i > 0, j ≥ 0
• Sn (2i) ⋄ Sn (2j) ≡W Sn (2i + 2j)
• Sn (2i + 1) ⋄ Sn (2j) ≡W Sn (2i + 2j + 1)
• Sn (2i) ⋄ Pn (2j) ≡W Pn (2i + 2j)
• Sn (2i + 1) ⋄ Pn (2j) ≡W Pn (2i + 2j)
• Pn (2i) ⋄ Sn (2j) ≡W Sn (2i + 2j)
• Pn (2i + 1) ⋄ Sn (2j) ≡W Pn (2i + 2j + 1)
• Pn (2i) ⋄ Pn (2j) ≡W Pn (2i + 2j − 2)
• Pn (2i + 1) ⋄ Pn (2j) ≡W Pn (2i + 2j)
• Sn (2i + 1) ⋄ Sn (2j + 1) ≡W Sn (2i + 2j + 1)
• Sn (2i + 1) ⋄ Pn (2j + 1) ≡W Pn (2i + 2j + 2)
• Pn (2i + 1) ⋄ Sn (2j + 1) ≡W Pn (2i + 2j + 2)
• Pn (2i + 1) ⋄ Pn (2j + 1) ≡W Pn (2i + 2j + 1)
The equivalences above, together with closure by ⋄, immediately provide complete LGA-recognizable languages for Dk (Σ0n ) for each k, n. Building upon this
we produce the whole Wadge hierarchy of LGA-recognizable languages.
6.5.2
Bestiarum vocabulum
For an ordinal α let exp(α) = ω1α . Hence,
α
ω1
··
k+1
exp
k
(α) = exp(exp (α)) =
ω·
ω1 1
| {z }
.
k+1 times ω1
To ease notation, we sometimes assume that exp0 (α) = 1.
Before describing the hierarchy, we recall the Wadge degrees of Dk (Σ0n )complete sets.
136
CHAPTER 6.
Proposition 6.14 ([48]). For each k > 0, dW (Sn (k)) = dW (Pn (k)) = expn (k).
The aim of the remaining part of this chapter is to provide an effective
characterization of the Wadge hierarchy for linear game automata (LGA Wadge
hierarchy for short). In particular we want to prove that linear game automata
P
recognize only languages of levels β = 0i=n βi , where each βi is of the form
βi = expi (ω)ηi +
1
X
expi (p)kp
p=ji
with ηi < ω ω , k2q ∈ {0, 1}, and ji , k2q+1 < ω. In order to verify this statement
we therefore have to check two points:
(1) the class of ordinals β is closed under the ordinal counterpart of the action
of a state (upper bound),
(2) every level of degree β is inhabited by an LGA recognizable language (lower
bound).
As a start, we provide for every such ordinal β an automaton Aβ , such that
L(Aβ ) is non self dual. In Theorem 6.18, we will verify that dW (L(Aβ )) = β.
To make the notation more readable, we use bracketed ordinal [β] to denote
the automaton Aβ . Since LGA are closed under complementation, when we
construct an automaton recognizing a non self dual set of degree β, we also
immediately get the automaton [β]∁ . We write [β]± for [β] ∨ [β]∁ .
Let us start with the basic building bricks of our construction: the automata
[1], [ω m ], [expi (1)], and [expi (ω)ω p ]. Together with these automata we show how
to make a step with those ordinals, i.e., how to define the automaton for [α + γ],
once we already have the automaton [α] and γ is one of the above. Let
[1] = ⊥ ,
[α + 1] = h−i(⊥, [α]± ) .
Note that [2] = K1Σ , and [2]∁ = K1Π . For m > 1 let
[ω] = [−]([3]∁ , ⊤) ,
[ω m ] = [−]([ω m−1 + 1]∁ , ⊤) ,
[α + ω] = [−]([3]∁ , [α + 1]∁ ) ,
[α + ω m ] = [−]([ω m−1 + 1]∁ , [α + 1]∁ ) .
For i > 1 let
[exp(1)] = h−i([2]∁ , ⊥) ,
[expi (1)] = h−i([expi−1 (1)]∁ , ⊥) ,
[α + exp(1)] = h−i([2]∁ , [α + 1]) ,
[α + expi (1)] = h−i([expi−1 (1)]∁ , [α + 1]) .
Σ
Π
Note that [expi (1)] = Ki+1
, [expi (1)]∁ ≡ Ki+1
. For p > 0 let
[expi (ω)] = [−]([expi (2)]∁ , ⊤) ,
[α + expi (ω)] = [−]([expi (2)]∁ , [α + 1]∁ ) ,
[expi (ω)ω p ] = [−]([expi (ω)ω p−1 + 1]∁ , ⊤) ,
[α + expi (ω)ω p ] = [−]([expi (ω)ω p−1 + 1]∁ , [α + 1]∁ ) .
6.5. THE WADGE HIERARCHY OF LINEAR GAME AUTOMATA
137
Thus, from the player’s point of view when involved in Wadge Games, a player
in charge of, for example, the language recognized by [expi (ω)] is given the
choice between the languages recognized by the [expi (n)]∁ ’s. The decision is
determined by the number of labels a played on the leftmost branch of the
tree before the first b. If the player keeps not playing b forever on the leftmost
branch, the tree will be rejected.
Using the basic buildingPblocks and basic steps defined above wea can in1
ductively define automata [ i=n δi ], such that each δi is of the form expi (ω)η +
i
ω
exp (1)p with η < ω and p < ω.
To define automata for all β described above, we need one more kind of
bricks and two more kinds of steps. For η < ω ω , 1 ≤ i < ω, we have:
[expi (2)] = [expi (1)] ⋄ [expi (1)]∁
[α + expi (ω)η +
1
X
expi (p + 2)kp ] = [α + expi (ω)η +
p=m
[α+expi (ω)η+
1
X
expi (p)kp ] ⋄ [expi (2)]
p=m
expi (p+2)kp +expi (2)] = [α+expi (ω)η+
p=m
6.5.3
1
X
ℓ
X
expi (p)kp +1]⋄[expi (2)] .
p=m
Computing Wadge degrees
The strategy we use in order to give an effective description of the LGA Wadge
hierarchy is as follows. We start by defining the notion of signed ordinal. Intuitively, ordinals with sign “+” denote Wadge degrees of non self-dual sets such
that, in Wadge games, the player in charge of one of those sets is accepting at
the beginning of any play. Dually, ordinals with sign “−” denote Wadge degrees
of non self-dual sets such that, in Wadge games, the player in charge of one of
those sets is rejecting at the beginning of any play. Ordinals with sign “±”
denote instead Wadge degrees of self dual sets. Then,P
we construct canonical
1
sets for every signed degree of the form expi (ω)ηi + p=ji expi (p)ηi′ + · · · +
P1
exp(ω)η1 + p=j1 exp(p)η1′ + η0 . With their help we prove that the class Φ of
signed ordinals from the previous subsection is closed under some set-theoretic
operations and that the degree of the results of these operations can be computed. Thanks to this closure property, by verifying that every action of a state
corresponds to one of those operations, we are able to show that every language
recognized by a linear game automaton has a signed degree which belongs to Φ.
By checking that every canonical automaton Aβ has Wadge degree β, we finally
prove that linear game automata recognize exactly languages of levels in Φ. We
end the section by showing that the problem of determining the signed degree
of any LGA recognizable language is decidable.
As expected, the technical difficulty of the effective description of the Wadge
hierarchy of linear game automata lies in the calculation of the Wadge degree
of sets of the form U ⋄ V , U ∨ V , supǫk (U hki ⋄ V ), for ǫ ∈ {+, −}, and U → V , if
dW (U ) = expi (1) for some i < ω. The calculations for the diamond operation
are used in the proof of Theorem 6.18. All of them are crucial for the proof of
Theorem 6.15. Their long description can be found in the Appendix.
A signed ordinal is a pair (ǫ, γ), where γ is an ordinal and ǫ ∈ {+, −, ±}.
Sometimes we call ǫ the sign of the ordinal γ. Slighty abusing notation, we write
138
CHAPTER 6.
[γ]ǫ for (ǫ, γ). For every γ =
P0
i=n
γi , where each γi is of the form
γi = expi (ω)ηi +
1
X
expi (p)ηi′
p=ji
with ηi , ηi′ < ω ω and ji < ω, for each possible sign ǫ we inductively define
canonical sets of trees C[γ]ǫ verifying the three following properties:
(1) dW (C[γ]ǫ ) = γ,
(2) (C[γ]+ )∁ ≡W C[γ]− ,
(3) C[γ]± ≡W C[γ]− ∨ C[γ]+ .
We thence say that a language L has signed degree [γ]ǫ if there is a canonical
C[γ]ǫ such that L ≡W C[γ]ǫ .
Let us start with the basic building bricks of our construction: the sets C[1]− ,
C[ωm ]− , C[expi (r)ωp ]− , and C[expi (ω)ωp ]− . Together with these sets we define the
canonical set C[α+γ]− of signed degree [α + γ]− as the set C[α]− + C[γ]− , once
we already have the sets C[α]− and C[γ]− . The canonical C[α+γ]+ and C[α+γ]±
are then defined as (C[α+γ]− )∁ and C[α+γ]− ∨ C[α+γ]+ respectively. Recall that
by T{a,b} we denote the space of all full binary trees over {a, b}. Let
C[1]− = ∅ ,
C[1]+ = T{a,b} ,
C[1]± = C[1]− ∨ C[1]+ .
For every i, p > 0, we take for C[expi (p)]− any Dp (Σ0i )-complete subset of T{a,b} .
Then, we define C[expi (p)]+ as (C[expi (p)]− )∁ , and C[expi (p)]± as C[expi (p)]− ∨
C[expi (p)]+ . For m > 1 let
• C[ω]− = sup−
k C[k]− ,
• C[ω]+ = (C[ω]− )∁ ,
• C[ω]± = C[ω]− ∨ C[ω]+ ,
hki
• C[ωm ]− = sup−
,
k (C[ω m−1 +1]+ )
• C[ωm ]+ = (C[ωm ]− )∁ ,
• C[ωm ]± = C[ωm ]− ∨ C[ωm ]− .
Analogously, for p > 1 and i > 0, let
• C[expi (ω)]− = sup−
k C[expi (k)]− ,
• C[expi (ω)]+ = (C[expi (ω)]− )∁ ,
• C[expi (ω)]± = C[expi (ω)]− ∨ C[expi (ω)]+ ,
hki
• C[expi (ω)ωp ]− = sup−
,
k (C[expi (ω)ω p−1 +1]+ )
• C[expi (ω)ωm ]+ = (C[expi (ω)ωm ]− )∁ ,
• C[expi (ω)ωm ]± = C[expi (ω)ωm ]− ∨ C[expi (ω)ωm ]− ,
6.5. THE WADGE HIERARCHY OF LINEAR GAME AUTOMATA
139
and, for r > 0 and i > 0,
hki
• C[expi (r)ωp ]− = sup−
,
k (C[expi (r)ω p−1 +1]+ )
• C[expi (r)ωm ]+ = (C[expi (r)ωm ]− )∁ ,
• C[expi (r)ωm ]± = C[expi (r)ωm ]− ∨ C[expi (r)ωm ]− .
Observe that whenever ǫ ∈ {+, −} all canonical C[γ]ǫ are non self-dual, and
self-dual when ǫ = ±. Moreover, as expected, their Wadge degree is exactly γ.
Roughly, assigning “+” to the signed degree of a canonical set C corresponds to
the fact that, from the player point of view in a Wadge Game, when the player
in charge of C starts to play he is accepting. Dually for “−”. P
0
Let Φ denote the set of signed ordinals of the form [β]ǫ = [ i=n βi ]ǫ , where
each βi is of the form
βi = expi (ω)ηi +
1
X
expi (p)kp
p=ji
with ηi < ω ω , k2q ∈ {0, 1}, and ji , k2q+1 < ω. Notice that for every [β]ǫ ∈ Φ
there is a canonical set C[β]ǫ .
The long combinatorial proof of the following effective closure property can
be found in the Appendix.
Theorem 6.15. For each γ1 , γ2 ∈ Φ it holds that the signed degree of Cγ1 ⋄ Cγ2 ,
hki
hki
−
Cγ1 ∨ Cγ2 , sup+
k (Cγ1 ⋄ Cγ2 ) and supk (Cγ1 ⋄ Cγ2 ) belong to Φ and their Wadge
degrees can be computed effectively. The same holds for Cγ1 → Cγ2 , if γ1 =
[expi (1)]ǫ for some i < ω and ǫ ∈ {+, −}.
In the next proposition we verify that every language recognized by a linear
game automaton has a signed degree which belongs to Φ. This is done essentially
by verifying that every action of a node corresponds, in set theoretic terms, to
one of the operations of Theorem 6.15.
Proposition 6.16. If L is LGA recognizable, then its signed degree belongs to
Φ.
Proof. In order to prove the proposition, we have to verify that if the languages
recognized by the automata A, B0 , B1 have signed degrees in Φ, then for every
state q, the signed degree of the automaton resulting from the action of q over
A, B0 , B1 still belongs to Φ. Because of Lemma 6.1, we consider linear game
automata in normal form over the alphabet {a, b}. The proof goes by induction
on the number of states. For the base case, assume the considered automaton
has only a single all rejecting, resp. all accepting state. Its signed Wadge degree
is then [1]− , resp. [1]+ . For the induction step, assume that the automaton is
obtained by the action of a existential state q. The universal case is obtained by
complementation. An existential state q can either be looping or not looping.
If it is not looping, than it is easy to see that q, in set-theoretic terms, is
acting either as an alternative (∨) or as a disjunctive product (⋄), or as a finite
combination of these two operations. Therefore, by Theorem 6.15 and induction
hypothesis it has a signed degree in Φ. If the considered node q is looping, two
situations may occur depending on the parity of q. If its parity is even, then the
140
CHAPTER 6.
node is acting like a supremum. More precisely, suppose that the result of the
action of q is the automaton h+i(A, B0 , B1 ). Then the language recognized by
hki
this automaton is Wadge equivalent to the set sup+
⋄ (L(B0 ) ⋄ L(B1 ))),
k (L(A)
and by Theorem 6.15 and induction hypothesis its signed degree belongs to Φ.
If the parity of q is odd, the result of the action of q is h−i(A, B0 , B1 ). In this
case we have that the language recognized by the resulting automaton is Wadge
equivalent to
−
L(h−i(A, ⊥)) → sup(L(A)hki ⋄ (L(B0 ) ⋄ L(B1 ))).
k
Two cases have to be scrutinized: (1) either L(A) is Σ0n -complete, or (2) L(A) ∈
∆0n+1 \ Σ0n , for a certain n > 0. In case (1), by Lemma 6.8, L(h−i(A, ⊥)) is still
Σ0n -complete and its signed degree is [expn 1]− . In case (2), by Theorem 6.9 and
Theorem 6.11, the language recognized by L(h−i(A, ⊥)) is Σ0n+1 -complete and
its signed degree is [expn+1 1]− . In both situations, by applying Theorem 6.15
and the induction hypothesis, the signed degree of the language recognized by
h−i(A, B0 , B1 ) belongs to Φ. Notice in particular that the previous argument
shows that, up to Wadge equivalence, unrestrained existential negative states
generate sets of the form C[expi (1)]− → C[γ]ǫ , with [γ]ǫ ∈ Φ.
From the previous theorem we immediately obtain that:
Corollary 6.17. The LGA Wadge hierarchy has height at most (ω ω )ω .
If we verify that the family of LGA-recognizable languages contains languages
L with dW (L) = β for every [β]ǫ ∈ Φ, we are able to prove that (1) linear game
automata recognize only languages of levels β, with [β]ǫ ∈ Φ, and that (2) the
height of the LGA hierarchy is exactly (ω ω )ω . This is done by showing that
canonical automata [β] are aptly named.
P0
Theorem 6.18. For every LGA automaton [β], dW ([β]) = β, where β = i=n βi ,
and each βi is of the form
βi = expi (ω)ηi +
1
X
expi (p)kp
p=ji
with ηi < ω ω , k2q ∈ {0, 1}, and ji , k2q+1 < ω.
Proof. The proof goes by induction on such ordinals. The initial case [1] is
trivial. For the induction step, we have to consider all the possible building
blocks described in Subsection 6.5.2.
First remark that, by applying the inductive hypothesis and the definition of
the set-theoretic operations of sum and of multiplication by ω, the automaton
[α + 1] has Wadge degree α + 1, and the automata [ω m ] and [expi (ω)ω p ] have
respectively Wadge degree ω m and expi (ω)ω p , for p, m ≥ 1. Since [expi (1)] =
Σ
Ki+1
, by Propositions 6.2 and 6.14 we obtain that its Wadge degree is exactly expi (1). By applying Lemma 6.13 and Proposition 6.14, we also obtain
that the Wadge degree of [expi (2)] is expi (2) and that dW (L([expi (ω)])) =
supn∈ω (expi (n)) = expi (ω).
In order to verify the remaining cases, we reason as follows. First observe
that in Wadge Games, as soon as the player in charge of the language recognized
6.5. THE WADGE HIERARCHY OF LINEAR GAME AUTOMATA
141
by the automaton [α + expi (ω)]∁ , resp. [α + ω]∁ has played the first b on the
leftmost branch,he is like a player in charge of the set L([expi (p)] ⋄ [α + 1]), resp.
L([p] ⋄ [α + 1]), for a certain finite p. On the other hand, for a player in charge in
a Wadge Game of the language recognized by the automaton [α + expi (ω)ω p ]∁ ,
resp. [α + ω p ]∁ , as soon as he has played the first b on the leftmost branch,
he is like a player in charge of the set recognized by [expi (ω)ω p−1 n] ⋄ [α + 1],
resp. by [ω p−1 n] ⋄ [α + 1], for a certain finite n. Analogously, as long as the
player in charge of the language recognized by the automaton [α+expi (1)], resp.
[α + exp(1)], does not play any node labelled by b on the leftmost branch, he is
like a player in charge of the set L([expi (1)]), resp. L([exp(1)]). But as soon as
he plays the first b on the leftmost branch, he is like a player in charge of the
set L([expi−1 (1)]∁ ⋄ [α + 1]), resp. L([2]∁ ⋄ [α + 1]). Therefore, if we prove that
the following identities hold:
• dW (L([2]∁ ⋄ [α + 1])) = α + 2
• dW (L([expi−1 (1)]∁ ⋄ [α + 1])) = α + expi−1 (1)
• dW (L([ω p−1 n] ⋄ [α + 1])) = α + ω p−1 n,
• dW (L([expi (ω)ω p−1 n] ⋄ [α + 1])) = α + expi (ω)ω p−1 n,
we are done for the considered cases. But those identities can be verified by
applying the computations for the diamond operation presented in Section A.3
of the Appendix. For the same reason, we obtain that the
degree of the
PWadge
1
language recognized by the automaton [α + expi (ω)η + p=m expi (p + 2)kp +
P1
expi (2)] has degree α + expi (ω)η + p=m expi (p + 2)kp + expi (2), and that the
P1
language recognized by [α + expi (ω)η + p=m expi (p + 2)kp ] has exactly the
P
degree α + expi (ω)η + 1p=m expi (p + 2)kp .
Observe that, by analyzing the proof of the previous theorem, it can be
easily verified that the signed Wadge degree of every automaton [β] is [β]− ,
the signed Wadge degree of every automaton [β]∁ is [β]+ , and that the signed
Wadge degree of every automaton [β]± is [β]± .
As a corollary we obtain the expected lower bound on the height of the
hierarchy.
Corollary 6.19. The LGA Wadge hierarchy has height at least (ω ω )ω .
From Corollaries 6.17 and 6.19, we finally have that:
Proposition 6.20. The Wadge hierarchy of linear game automata has the height
2
(ω ω )ω = ω ω .
We end this section by showing that the problem of determining the signed
Wadge degree of a LGA-recognizable tree language is decidable.
Theorem 6.21. For each LGA we can calculate effectively the signed degree of
the recognized language.
Proof. We proceed by induction on the number of states. Let C be an LGA.
If C has only one state, it is either totally accepting or totally rejecting. In
the first case the signed degree is [1]+ , in the second case it is [1]− . Suppose
142
CHAPTER 6.
that C has more states. By duality we may assume that the initial state q0 is
existential: if it is universal, compute the signed degree for the complement of
C, and return the degree negated. Suppose that q0 is not looping. By linearity,
C can be represented as in the figure below for some automata A0 , A1 , B0 , B1 ,
each having less states than C.
A0 o
a,0
h1i
}}
}}
}
}
}~ }
b,0
B0
a,1
/ A1
AA
AA b,1
AA
AA
B1
Clearly L(C) ≡ L(A0 ) ⋄ L(A1 ) ∨ L(B0 ) ⋄ L(B1 ). Hence, we can use the induction
hypothesis to get the degrees of L(Cqi ), and then Theorem 6.15 to compute
dW (C) = dW (Cq1 ) ⋄ dW (Cq2 ) ∨ dW (Cq3 ) ⋄ dW (Cq4 ).
If q0 is looping, we can assume w.l.o.g. that C is of the form shown in the
figure below with i = 0, 1.
a,0
+ hii a,1 / A
AA
}
AAb,1
b,0 }}
AA
}}
}
A
~}
B0
B1
If i = 1, there exists n ∈ ω such that L(A) is either Σ0n -complete, or in ∆0n+1 \Σ0n .
If L(A) is Σ0n -complete, by Lemma 6.8, the language recognized by C ′ , defined
in the figure below is also Σ0n -complete.
a,0
* h1i
a,1
/A
b,∗
⊥
Since dW (A) = dW (C ′ ) = [expn (1)]− and ([expn (1)]− )hki = [expn (1)]− for
each k > 0, we have dW (C) = [expn (1)]− → dW (B1 ) ⋄ dW (B2 ) ⋄ [expn (1)]− .
On the other hand, if L(A) ∈ ∆0n+1 \ Σ0n , by Theorem 6.9 and Theorem 6.11,
the language recognized by C ′ is Σ0n+1 -complete, and we already observed that
hki
dW (C) = [expn+1 (1)]− → dW (sup−
⋄ (B1 ⋄ B2 )). We conclude by the
k (A
inductive hypothesis and Theorem 6.15.
hki
If i = 0, dW (C) = sup+
⋄dW (B1 ⋄B2 )), and again the claim follows
k (dW (A)
from Theorem 6.15 and the induction hypothesis.
6.6
Summarizing remarks
While for ω-regular languages the understanding of the corresponding index and
topological hierarchies is complete, for trees the situation is not so satisfactory.
The only case examined satisfactorily is that of deterministic automata. This
is due to the work of Niwinski and Walukiewicz [101, 102], and of Murlak [90,
6.6. SUMMARIZING REMARKS
143
92, 93]. For non-deterministic or alternating automata the only results obtained
are strictness theorems for various classes [32, 33, 89, 96], and lower bounds for
the heights of the hierarchies [51, 114].
In this chapter we proposed a novel class of automata, named linear game
automata, capturing an interesting aspect of alternation and with all three hierarchies – index, Borel, and Wadge – decidable. Moreover we shown that the
weak index and the Borel rank coincide over LGA-recognizable languages.
We saw that, despite their apparent simplicity, LGA yield a class of languages
surprisingly complex from the topological point of view: the height of their
Wadge hierarchy is (ω ω )ω . Admittedly, this is much less than the hierarchy for
weak alternating automata, which is known to be at least ǫ0 high [51]. But
this was expected, as LGA form a very restricted subclass of weak alternating
automata. What is surprising however, is that the height of the Wadge hierarchy
for LGA is much larger than that for deterministic automata, which was shown
in [92] to be (ω ω )3 + 3, and the same as for deterministic push-down automata
on infinite words [49].
144
CHAPTER 6.
Conclusion
This thesis focused on the complexity of some fragments of the modal µ-calculus,
or equivalently of some subclasses of alternating automata. More precisely, in
the first part of this work we have studied the expressive power of the modal
µ-calculus over some restricted classes of transition systems, while in the second
part we gave a complete effective characterization of the three classical hierarchies (index, Borel and Wadge) for a class of tree automata capturing a very
weak form of alternation.
For what concerns the First Part of the dissertation, some of the results
obtained in Chapter 2 and Chapter 3 can be summarized by the following
figure:
KT
fixpoint
alternation
hierarchy
strict
K4
collapse to the
alternation
free fragment
KB4
collapse to the
modal
fragment
GL
collapse to the
modal
fragment
where KT stands for the class of all reflexive transition systems, K4 for the
class of all transitive transition systems, KB4 for the class of all transitive and
symmetric transition systems, and GL for the class of all transitive and upward
well-founded models. The collapse of the modal µ-calculus into its modal fragment over GL was already proved by van Benthem in [13] and Visser in [123]
by using the De Jongh-Sambin fixpoint theorem. Our proof is independent of
this important result, and uses results from Chapter 2 and the fact that on
upward well-founded models the modal µ-calculus collapses into the first ambiguous class of the fixpoint alternation hierarchy. In extending the language
by allowing fixpoints to bound also negative occurrences of free variables and
showing that it collapses on the modal fragment, we were then able to provide
a new proof of the uniqueness theorem of Bernardi, De Jongh and Sambin and
a constructive proof of the existence theorem of De Jongh and Sambin.
Chapter 4, which is the last chapter of the first part, can be seen as a kind
of “bridge” between the first and the second parts of the thesis. Indeed, in this
chapter we shown that on transitive models modal logic corresponds exactly to
the Borel fragment of the modal µ-calculus. This was done by providing a bunch
of equivalent effective characterizations for the temporal logic EF on arbitrary
trees. More specifically we proved that up to EF-bisimilarity, the property of
being definable by an EF formula and the property of being a Borel set coincide.
Since we were able to verify that every language definable in weak monadic second order logic with the child relation is Borel, we immediately obtained that
the logic EF is the EF-bisimulation invariant fragment of WMSO. By verifying
145
146
CONCLUSION
that all these properties are also equivalent with an effective algebraic characterization of EF-definability for finitely branching trees given by Bojanczyk and
Idziaszek [24], as a corollary we obtained their decidability.
In the much shorter Second Part of the dissertation we introduced a new
subclass of weak alternating tree automata, linear game automata, and provided an effective characterization for all the three corresponding hierarchies:
index hierarchy, Borel hierarchy and Wadge hierarchy. Moreover, we verified
that for every language recognized by those automata, the Borel rank and the
Mostowski-Rabin index coincide, making another step towards a positive answer
to Murlak’s conjecture, stating that for weakly recognizable tree languages the
weak index hierarchy and the Borel hierarchy coincide level by level ([93]).
Future work
Part 1
Concerning the modal µ-calculus hierarchy on restricted class of models, the
missing case is the symmetric case. It would be nice to fill in this gap. We
conjecture that the fixpoint hierarchy is strict for this class of transition systems.
Another open problem is to decide for any given definable language its position in the fixpoint hierarchy. This is a very difficult and important question.
To our knowledge, it is only known how to decide the low levels of the hierarchy
([78, 103, 129]). As we saw, recently, Colcombet and Löding [42] have been able
to reduce the analogous problem for tree automata to the uniform universality
problem for what are called distance-parity automata.
Also, finding new, possibly effective, characterizations of tree logics is another important problem in the area. As an extension of a result presented in
this dissertation, it would be for instance nice to know whether on full trees
it holds that any recognizable tree language is Borel iff it is definable in weak
monadic second order logic.
Finally note that, although Janin and Lenzi [63] proved1 that over the class of
graphs of bounded degree, every µ-formula can be translated into a MSO formula
of the third monadic class Σ3 , the question whether the modal µ-calculus is
equivalent to the bisimulation invariant fragment of some fixed level of the
quantifier alternation hierarchy of MSO remains open.
Part 2
We believe that the notion of game automata is well suited to take us further.
Indeed, the next step is to consider weak and then strong game automata, where
weak strong automata are weak alternating automaton where the transition
relation is a total function, while strong game automata are (strong) alternating
automaton where the transition relation is a total function. This last class is
already quite expressive, as it contains inhabitants of every level of the (strong)
index hierarchy and subsumes deterministic languages. Extending decidability
of the index, Borel/projective and Wadge hierarchies to this class would be an
1 The authors remark that their result is a consequence of the work of Courcelle [43], who
shows that this is true on a quite general class of graphs.
147
important result, though possibly the last one accessible with the tools we used
for instance in Chapter 6.
Because of its difficulty, the general case probably requires new techniques.
However for a start, it would be very nice to already solve those hierarchical
questions for weak alternating automaton. A first step in this direction would
be finding an answer to Murlak’s conjecture by showing that for weakly recognizable languages, the weak index and the Borel rank coincide. Another step
would be also to prove that the lower bound obtained by Duparc and Murlak
for the corresponding Wadge hierarchy is tight.
Nowadays, there is a regain of interest among the automata theory community in the idea of considering quantitative extensions of the standard theory
of regular languages. This is due to a series of papers of Bojanczyk, Colcombet, Kirsten and Löding, among others. First of all, Kirsten [72] gave a much
simpler and self-contained proof of the one proposed by Hashiguchi [58, 59] for
the star-height decision problem through a reduction to the limitedness problem for a form of automata called nested distance desert automata2 . The same
strategy was used by Colcombet and Löding. In [41], the star-height problem
over trees has been solved by a reduction to the limitedness problem of nested
distance desert automata over trees. As we already mentioned, in [42] they
tried a similar attempt for solving the problem of deciding the index hierarchy
of non-deterministic automata over infinite trees.
At the same time as Kirsten’s work, Bojanczyk [19] introduced an extension MSOU of MSO, where a new second-order quantifier UX.ψ(X) was added
meaning “there exists a set X (of infinite trees or words) as big as I want such
that ψ(X) holds”. He showed that for two fragments of this logic the satisfiability problem is decidable. The word cases were studied in more depth by
Bojanczyk and Colcombet [23]. In this paper, by introducing BS-automata, a
certain class of counter non-deterministic automata very close to distance desert
automata, the authors provided some fragments of MSOU that have decidable
satisfiability over infinite words. But non-determinism is important for full
MSO, where existential quantication over infinite sets is allowed, and it comes
with a cost: BS-automata are not closed under complementation, and it is not
clear what the correct automaton model for full MSOU is. Moreover, it is still
an open problem if full MSOU has decidable satisfiability over infinite words.
Interestingly, all these works point to the following argument: there are robust
extensions of regular languages, extensions that have descriptions in terms of
both automata and logic. This argument was investigated in two recent papers
by Bojanczyk [21, 25], the last one joint with Torunczyk, where new classes of
languages of infinite words were defined and were showed to have two equivalent descriptions: in terms of a deterministic counter automaton (called a Max-,
resp. Min-automaton), and in terms of an extension of weak MSO. Moreover, it
was shown that Min- and Max-automata fit in a more general picture, where deterministic automata with prefix-closed acceptance conditions define extensions
of weak MSO.
The study of those quantitative extensions of regular languages appears to
be a particularly fertile and interesting territory for the use of topological meth2 A distance desert automaton is a non-deterministic automaton running over words which
can count the number of occurrences of some “special” states. A nested distance desert
automaton is nothing but a distance desert automaton in which multiple counters and reset
of the counters are allowed (with a certain constraint of nesting of counters).
148
CONCLUSION
ods. Already in [21], Bojanczyk used a topological argument in order to show
that non-deterministic Max-automata recognize strictly more languages than
deterministic ones. Cabessa, Duparc, Facchini and Murlak [38] compared ωregular and Max-regular languages in terms of topological complexity. The
authors proved that up to Wadge equivalence the classes coincide. Moreover,
when restricted to ∆02 -languages, they showed that the classes contain virtually
the same languages. On the other hand, separating examples of arbitrary complexity exceeding ∆02 were constructed. More recently, Hummel, Skrzypczak
and Torunczyk [62] were able to show that MSOU on ω-words can even define non-Borel sets. From this fact, they concluded that there is no model of
non-deterministic automata with a Borel acceptance condition which captures
all of MSOU. In the same article, the authors also gave an exact topological
complexity of the classes of languages recognized by non-deterministic B-, Sand BS-automata on infinite words. Furthermore, they verified that, since they
inhabit all finite levels of the Borel hierarchy, the corresponding alternating
automata have higher topological complexity than non-deterministic ones.
All the previous papers show that, as future work, it would be very interesting to further study the topological complexity of these quantitative extensions
of regular languages. It would be for instance nice to have an effective description of the Wadge hierarchy of deterministic and non-deterministic counter
automata, but also to know whether there are non Borel languages recognized
by an alternating BS-automaton.
149
150
CONCLUSION
Appendix A
Computations of Chapter 6
A.1
Basic properties of operations
Remember that we say that a non self dual set L is initializable if L ≥W L → L. The following lemmas
summarize simple yet useful properties of the operations on languages. They can be proved with standard
Wadge game arguments.
Lemma A.1. For initializable A, B and arbitrary A′ , B ′ and An , Bn , n < ω
(A → A′ ) ⋄ (B → B ′ ) ≡W A ⋄ B → ((A → A′ ) ⋄ B ′ ∨ A′ ⋄ (B → B ′ )) ,
(A → A′ ) ⋄ B ≡W A ⋄ B → A′ ⋄ B ,
+
+
(A → A′ ) ⋄ sup Bn ≡W sup (A → A′ ) ⋄ Bn ,
n
n
−
−
−
+
′
′
′
′
(A → A ) ⋄ sup Bn ≡W A → (A ⋄ sup Bn ) ∨ (sup (A → A ) ⋄ Bn ) ∨ (sup (A → A ) ⋄ Bn )
n
n
n
n
−
−
−
(⊥ → A′ ) ⋄ sup Bn ≡W ⊥ → (A′ ⋄ sup Bn ) ∨ (sup (⊥ → A′ ) ⋄ Bn ) ,
n
n
n
−
−
+
′
′
′
(⊤ → A ) ⋄ sup Bn ≡W ⊤ → (A ⋄ sup Bn ) ∨ (sup (⊤ → A ) ⋄ Bn ) ,
n
n
n
+
+
+
+
+
+
sup Am ⋄ sup Bn ≡W ⊤ → sup(Am ⋄ sup Bn ) ∨ (sup (sup Am ) ⋄ Bn ) ,
m
n
m
n
n
m
+
−
+
−
+
+
sup Am ⋄ sup Bn ≡W ⊤ → sup(Am ⋄ sup Bn ) ∨ (sup (sup Am ) ⋄ Bn ) ,
m
n
m
n
n
m
−
−
−
−
−
−
sup Am ⋄ sup Bn ≡W ⊥ → sup(Am ⋄ sup Bn ) ∨ (sup (sup Am ) ⋄ Bn ) .
m
n
m
n
n
m
Lemma A.2. For arbitrary sets A, B, C, D it holds that
• (A ∨ B) ⋄ C ≡W A ⋄ C ∨ B ⋄ C,
• if A ≤W C and B ≤W D, then A ⋄ B ≤W C ⋄ D.
Let us now recall the elegant relation between operations on sets and Wadge degrees.
151
for A > ⊥ ,
152
Computations of Chapter 6
Proposition A.3 ([48]). For arbitrary Borel sets L, M, Ln with n < ω it holds that
dW (M + L) = dW (M ) + dW (L) ,
−
+
n
n
dW (sup Ln ) = dW (sup Ln ) = sup(dW (Ln ) + 1) .
A.2
n
Closure by ∨ and →
µ
n
for µ ∈ {+, −, ±}
For n > 0, let Φ̂n denote the set of signed ordinals of the form ΣN
i=1 exp (i)ki
and some natural N , ki , with ki ∈ {0, 1} for even i. For uniformity, Φ̂0 = {0+ , 0− , 0± }. Let Φn =
µ
µ
{[expn (ω)α + β] : µ ∈ {+, −, ±} , α < ω ω , β µ ∈ Φ̂n } for n > 0 and Φ0 = {[α] : µ ∈ {+, −, ±} , α < ω ω }.
µ
Recall that by Φ we denote the set of signed ordinals of the form [αk + . . . + α1 + α0 ] , with α+
i ∈ Φi .
µ
To make the notation more readable, in what follows we write [α] for the canonical set C[α]µ of signed
degree [α]µ .
The closure of Φ by ∨ is very simple. If [α]µ and [β]ν are comparable, [α]µ ∨ [β]ν is simply equal to the
larger of the two. If [α]µ and [β]ν are incomparable, then necessarily [β]ν = [α]µ̄ and the result is [α]± .
Let us now concentrate on →. First we state yet another simple observation. By µ̄ we denote the dual sign:


+ if µ = − ,
µ̄ = − if µ = + ,


± if µ = ± .
Lemma A.4. For arbitrary tree sets An ,
(
supνn An
ν
µ
[1] → sup An ≡w
−
[1]µ → (sup+
n
n An ) ∨ (supn An )
if µ = ν ∈ {−, +}
if µ = ν̄ ∈ {−, +} .
Observe that by Proposition A.3 [expi (1)]+ → [β]± ≡W [β + expi (1)]+ . Thus, it follows that the result is
in Φ. Consider now [expi (1)]+ → [β]ν for ν ∈ {−, +}. If [β]ν = B → [β ′ ]± , with B initializable, it follows
that [expi (1)]µ → [β]ν ≡W ([expi (1)]+ → B) → [β ′ ]± . It is easy to see that

i
+

for B ≤W [expi (1)]+
[exp (1)]
[expi (1)]+ → B ≡W B
for B ≥w [expi (1)]+ ,


[expi (1)2]+ for B ≡W [expi (1)]− ,
and we can conclude from the previous case. The remaining case is that of [β]ν = supν [β ′ + βn ]+ , with
βn = expi (ω)ω p n or βn = expi (n). If i > 0, we have [expi (1)]µ → [β]ν ≡W ([expi (1)]µ → [1]ν̄ ) →
[β]ν ≡W [expi (1)]µ → ([1]ν̄ → [β]ν ) By Lemma A.4, this is equal to [expi (1)]µ → [1]ν̄ → [β]± . Hence,
[expi (1)]µ → [β]ν ≡W [expi (1)]µ → [β]± and we conclude by the first case. For i = 0 use Lemma A.4.
A.3
Closure by ⋄
First of all note that since β + ∨ β − = β ± , by Lemma A.2 αµ ⋄ β ± corresponds to (αµ ⋄ β + ) ∨ (αµ ⋄ β − ).
Therefore, if αµ ⋄ β ν ∈ Φ, for both ν = +, −, αµ ⋄ β ± ∈ Φ holds because we know that Φ is closed under ∨.
Lemma A.5. Fix a natural number n > 0. For each [α]µ , [β]ν ∈ Φ̂n , one can effectively find [γ1 ]λ1 , [γ2 ]λ2 , [γ3 ]λ3
such that [γi ]λi ∈ Φ̂n or [γi ]λi = [γi′ + 1]λ1 and
[α]µ ⋄ [β]ν = [γ1 ]λ1 ,
[α + 1]µ ⋄ [β]ν = [γ2 ]λ2 ,
[α + 1]µ ⋄ [β + 1]ν = [γ3 ]λ3 .
A.3. CLOSURE BY ⋄
153
Proof. We only give a proof of the first assertion. The remaining two are very similar.
We proceed by induction on the sum of the coefficients of α and β. The basic step is α = expn (p),
β = expn (q). By Lemma 6.13, we find r and λ such that αµ ⋄ β ν = expn (r)λ .
In the inductive step we give argument for the case αµ = expn (p)µ → (γ − ∨ γ + ) and β ν = expn (q)ν →
(δ − ∨ δ + ); the case when α = expn (p) or β = expn (q) is very similar, only uses the second assertion of
Lemma A.1.
Since expn (p)µ and expn (q)ν are initializable, by Lemma A.1 and Lemma A.2, we get
αµ ⋄ β ν = expn (p)µ ⋄ expn (q)ν → αµ ⋄ δ − ∨ αµ ⋄ δ + ∨ γ − ⋄ β ν ∨ γ + ⋄ β ν .
By induction hypothesis we compute αµ ⋄ δ − , αµ ⋄ δ + , γ − ⋄ β ν , γ + ⋄ β ν ∈ Φ̂n and by the properties of ∨ we
get η κ = αµ ⋄ δ − ∨ αµ ⋄ δ + ∨ γ − ⋄ β ν ∨ γ + ⋄ β ν ∈ Φ̂n . Again by Lemma 6.13 we get an expression of the
λ′
P
1
n
form expn (r)λ → η κ , which can be presented as
. Indeed, by an argument analogous
i=N exp (i)mi
PL
n
to the one used after Lemma A.4, one shows that for η = i=M exp (i)ℓi with ℓL > 0 and for λ ∈ {+, −} ,


ηκ
if L > n and κ ∈ {+, −} , or





L = n and κ = λ ,

[expn (r)]λ → η κ ≡W [η + expn (r)]λ
if L ≥ n and κ = ± , or



L = n and κ = λ̄ ,



 [P r
n
n
λ
exp
(i)ℓ
+
exp
(r)]
if
L < n.
i
i=M
It remains to show that mi ∈ {0, 1} for even i.
Suppose first that r is even, and λ = −. Then by Lemma 6.13 p and q are even, r = p+q and µ = ν = −. For
each of αµ ⋄ δ − , αµ ⋄ δ + , γ − ⋄ β ν , γ + ⋄ β ν we can apply Lemma A.1, and get an expression expn (s)κ → ικ1 ,
where ικ1 can be again obtained from the inductive hypothesis. The possible values for expn (s)κ are
expn (p)− ⋄ expn (t)− = expn (p + t)− or expn (p)− ⋄ expn (t)+ = expn (p + t)+ with t > q (since q is even
and β ν ∈ Φ), or expn (q)− ⋄ expn (u)− = expn (q + u)− or expn (q)− ⋄ expn (u)+ = expn (q + u)+ with u > p
(since p is even and αµ ∈ Φ). Hence, s > p + q = r. In consequence, by the properties of → and ∨,
n
κ
η κ = (ΣM
i=L exp (i)ℓi ) with L > r, ℓL > 0. Hence,
!κ PM
M

X
expn (i)ℓi + expn (r)− κ = ±
i=L
κ
.
αµ ⋄ β ν = expn (r)− →
= PM
expn (i)ℓi
n

κ ∈ {−, +}
exp
(i)ℓ
i
i=L
i=L
In either case the result is in Φ̂n .
Next, suppose that r is even, and λ = +.


p + q
r = p+q−1


p+q−2
Then,
p ≡ q (mod 2), µ = ν̄
p odd, q even, ν = + (or symmetrically)
p, q even, µ = ν = +
Again, we can present each of αµ ⋄ δ − , αµ ⋄ δ + , γ − ⋄ β ν , γ + ⋄ β ν as expn (t)κ → ικ1 with ικ1 ∈ Φ̂n . In order
to carry on like before it is enough to show that in no case expn (t)κ = expn (r)− . Indeed, if we exclude this
possibility, by inductive hypothesis and by the properties of → and ∨, we can conclude that
P
κ
M
n

exp
(i)ℓ
i
i=L
η κ = PM
n
+
n

i=L exp (i)ℓi + exp (r)
154
Computations of Chapter 6
for L > r, ℓL > 0, and so
P
κ
M
n

exp
(i)ℓ
i
i=L
αµ ⋄ β ν = PM
n
+
n

i=L exp (i)ℓi + exp (r)
.
These two are both in Φ̂n , since η κ is. So it remains to see that expn (t)κ 6= expn (r)− . Observe that only an
expression of the form expn (2i)− ⋄ expn (2j)− can give expn (r)− for even r. The only case giving a chance
of such an expression is when r = p + q. But then 2i = p and, since β ν ∈ Φ, 2j > q (or symmetrically), so
2i + 2j > p + q = r, and we are safe.
If r is odd, the claim follows easily by induction hypothesis, and by properties of →.
Lemma A.6. Let N > 0 and [α0 ]µ , [β0 ]ν ∈ Φ̂N , with µ, ν ∈ {+, −}. Define α̂0 = α0 + εα , and β̂0 = β0 + εβ
with εα , εβ ∈ {0, 1}. Let α = expN (ω)(ω k ak +. . .+ωa1 +a0 )+ α̂0 and β = expN (ω)(ω k bk +. . .+ωb1 +b0 )+ β̂0 .
Let ℓα be the least i for which ai > 0, and similarly for ℓβ .
• If α0 > 0, β0 > 0, then
±
[α]µ ⋄ [β]ν = [α̂0 ]µ ⋄ [β̂0 ]ν → expN (ω)(ω k (ak + bk ) + . . . + ω(a1 + b1 ) + (a0 + b0 )) .
• If α0 > 0, β̂0 = 0, then
+
[α]µ ⋄ [β]+ = expN (ω)(ω k (ak + bk ) + . . . + ω ℓβ (aℓβ + bℓβ )) ,
±
[α]µ ⋄ [β]− = [α̂0 ]µ → expN (ω)(ω k (ak + bk ) + . . . + ω(a1 + b1 ) + (a0 + b0 )) .
• If α0 = 0, β0 = 0, then
−
[α]− ⋄ [β]− = expN (ω)(ω k (ak + bk ) + . . . + ω(a1 + b1 ) + (a0 + b0 ))
−
[α]− ⋄ [β]− = expN (ω)(ω k (ak + bk ) + . . . + ω(a1 + b1 ) + (a0 + b0 )) + 1
−
[α]− ⋄ [β]− = expN (ω)(ω k (ak + bk ) + . . . + ω(a1 + b1 ) + (a0 + b0 )) + 1
−
[α]− ⋄ [β]− = expN (ω)(ω k (ak + bk ) + . . . + ω(a1 + b1 ) + (a0 + b0 ))
+
[α]− ⋄ [β]+ = expN (ω)(ω k (ak + bk ) + . . . + ω ℓβ (aℓβ + bℓβ ))
+
[α]− ⋄ [β]+ = expN (ω)(ω k (ak + bk ) + . . . + ω(a1 + b1 ) + (a0 + b0 )) + 1
+
[α]+ ⋄ [β]+ = expN (ω)(ω k (ak + bk ) + . . . + ω ℓ (aℓ + bℓ − ε))
+
[α]+ ⋄ [β]+ = expN (ω)(ω k (ak + bk ) + . . . + ω ℓβ (aℓβ + bℓβ ))
for εα = εβ = 0,
for εα = εβ = 1,
for εα = 0 , εβ = 1 , ℓβ ≥ ℓα
for εα = 0 , εβ = 1 , ℓβ < ℓα
for εβ = 0 ,
for εα = 0 , εβ = 1 ,
for εα = εβ = 0 ,
for εα = 0 , εβ = 1 ,
where ℓ = max(ℓα , ℓβ ) and ε equals 1 if ℓα = ℓβ and 0 otherwise.
Proof. In the proof we assume that εα = εβ = 0, the remaining cases being very similar. We prove all the
P0
equations simultaneously by induction on (α, β). Let θ = expN (ω) i=k ω i (ai + bi ).
• Case α0 > 0, β0 > 0. Suppose α = α′ + expN (p), β = β ′ + expN (q). We have
[α]µ ⋄ [β]ν = [expN (p)]µ ⋄ [expN (q)]ν → [α]µ ⋄ [β ′ ]± ∨ [α′ ]± ⋄ [β]ν
Let α0 = γ + expN (p), β0 = δ + expN (p). We get
±
±
[α]µ ⋄ [β]ν = [expN (p)]µ ⋄ [expN (q)]ν → A → [θ] ∨ B → [θ]
= [expN (p)]µ ⋄ [expN (q)]ν → (A ∨ B) → [θ]±
= [α0 ]µ ⋄ [β0 ]ν → [θ]± .
with A = [α0 ]µ ⋄ [δ]± (or A = [α0 ]µ if δ = 0) and B = [γ]± ⋄ [β0 ]ν (or B = [β0 ]ν if γ = 0).
A.3. CLOSURE BY ⋄
155
• Case α0 > 0, β0 = 0. Suppose ℓβ > 0. Then by Proposition A.3 we get that [expN (ω)ω ℓβ ]+ =
N
ℓβ −1 +
sup+
i] . By the induction hypothesis for
i [exp (ω)ω
b′i
=
(
i 6= ℓβ
i = ℓβ
bi
bi − 1
we have



+
ℓβ
X
ℓ
−1
j
′
[α] ⋄ [β] = sup α ⋄ exp (ω) 
ω bj  + ω β i
+
+
µ
µ
i

+

= sup expN (ω) 
i

= expN (ω)
ℓβ
X
j=k
N
j=k
ℓβ
X
j=k

+
ω j (aj + b′j ) + ω ℓβ −1 (aℓβ −1 + i)
+
ω j (aj + bj )
.
N
+
Now, suppose that ℓβ = 0 (this also covers the induction basis). Then [expN (ω)]+ = sup+
i [exp (i)] , and
like before, using the case α0 > 0, β0 > 0,

+

[α]µ ⋄ [β]+ = sup αµ ⋄ expN (ω) 
i

j=k

+
ω j b′j  + expN (i)
± 
0
X
+
= sup αµ0 ⋄ expN (i) → expN (ω)
ω j (aj + b′j ) 
+
i

ℓβ
X


j=k
±
0
X
+
= expN (ω) → expN (ω) 
ω j (aj + b′j )+

= expN (ω)
ℓβ
X
j=k
j=k
+
ω j (aj + bj ) .
Let us move to the second equation. Let α = α′ + expN (p), β = β ′ + expN (ω)ω ℓβ . If ℓβ > 0, we have
µ
−
N
µ
[α] ⋄ [β] = [exp (p)] →
−
([α′ ]± ⋄ [β]− ) ∨ (sup [α]µ ⋄ [β ′ + expN (ω)ω ℓβ −1 n]+ )∨
n
+
µ
′
N
ℓβ −1 +
∨(sup [α] ⋄ [β + exp (ω)ω
n] ) .
n
h
P
i+
ℓβ
i
′
ℓβ −1
As [α]µ ⋄ [β ′ + expN (ω)ω ℓβ −1 n]+ = expN (ω)
(aℓβ −1 + n)
, we obtain
i=k ω (ai + bi ) + ω


[α]µ ⋄ [β]− =[expN (p)]µ → ([α′ ]± ⋄ [β]− ) ∨ expN (ω)
ℓβ
X
i=k
± 
ω i (ai + bi )  .
156
Computations of Chapter 6
If α′ = expN (ω)γ + δ, with [δ]± ∈ Φ̂N , δ > 0, then we conclude from the induction hypothesis that

± 
ℓβ
X
±
[α]µ ⋄ [β]− = [expN (p)]µ →  [δ]± → [θ] ∨ expN (ω)
ω i (ai + bi ) 

±
i=k
= [expN (p)]µ → [δ]± → [θ]
±
= [expN (p)]µ → [δ]± → [θ]
±
= [α0 ]µ → [θ] .
h
i+
Pα i
If α′ = expN (ω)γ, by induction hypothesis [α′ ]± ⋄ [β]− = [θ]− ∨ expN (ω) ℓi=k
ω (ai + bi ) , so

−
"
[α]µ ⋄ [β]− = [expN (p)]µ → [θ] ∨ expN (ω)
N
µ
= [exp (p)] → [θ]
±
ℓα
X
#+
ω i (ai + bi )
i=k

∨ expN (ω)
ℓβ
X
i=k
± 
ω i (ai + bi ) 
+
as one of the last two disjuncts must be at least [θ] . This concludes the case of ℓβ > 0.
If ℓβ = 0 we have
−
[α]µ ⋄ [β]− = [expN (p)]µ → ([α′ ]± ⋄ [β]− ) ∨ (sup [α]µ ⋄ [β ′ + expN (n)]+ )∨
n
+
∨(sup [α]µ ⋄ [β ′ + expN (n)]+ ) .
n
h
i±
P0
Since [α]µ ⋄ [β ′ + expN (n)]+ = [α0 ]µ ⋄ [expN (n)]+ → expN (ω) i=k ω i (ai + b′i ) , by Lemma A.5 we have
±
[α]µ ⋄ [β]− = [expN (p)]µ → ([α′ ]± ⋄ [β]− ) ∨ [θ]
If α′ = expN (ω)γ + δ, then α0 = δ + expN (p) and by induction hypothesis for [α′ ]± ⋄ [β]− we get
±
±
[α]µ ⋄ [β]− = [expN (p)]µ → (([δ]± → [θ] ) ∨ [θ] )
±
= [expN (p)]µ → [δ]± → [θ] )
= [α0 ]µ → [θ]± .
±
If α′ = expN (ω)γ, then α0 = expN (p) and by induction hypothesis [α′ ]± ⋄ [β]− is at most [θ] . Hence,
[α]µ ⋄ [β]− = [expN (p)]µ → [θ]
±
±
= [α0 ]µ → [θ] .
• Case α0 = 0, β0 = 0. Let α = α′ + expN (ω)ω ℓα , β = β ′ + expN (ω)ω ℓβ . Suppose that ℓα > 0, ℓβ > 0.
(n)
(n)
(n)
(n)
Set aℓα = aℓα − 1, aℓα −1 = n, and ai = ai for other i’s. Analogously define bi . Applying Lemma A.1
A.3. CLOSURE BY ⋄
157
and the inductive hypothesis we get the claim
[α]− ⋄ [β]− ) =
−
−
= ⊥ → sup [α]− ⋄ [β ′ + expN (ω)ω ℓβ −1 n]− ∨ sup [α′ + expN (ω)ω ℓα −1 n]− ⋄ [β]−
n
n

# 
"
#
"
−
0
0
−
X
X
−
−
(n)
(n)

ω i ai + b i
∨ sup expN (ω)
ω i ai + b i
= ⊥ → sup expN (ω)
n
= ⊥ → expN (ω)
"
n
i=k

N
= ⊥ → exp (ω)
−
ℓβ
X
ω i (ai + bi ) ∨ expN (ω)
i=k
0
X
i=k
"
#−
i
ω (ai + bi )
i=k
"
N
= exp (ω)
ℓα
X
#− 
i=k
0
X

ω i (ai + bi )
#−
i
ω (ai + bi )
i=k
,
[α]− ⋄ [β]+ =
+
+
′
N
ℓα −1 −
+
−
′
N
ℓβ −1 −
n] ⋄ [β]
n] ∨ sup [α + exp (ω)ω
= ⊤ → sup [α] ⋄ [β + exp (ω)ω

n
n
+
"
= ⊤ → sup expN (ω)
n

= ⊤ → expN (ω)

= ⊤ → expN (ω)
+
+
ℓβ
X
i=k
ℓβ
X
i=k
0
X
i=k
(n)
ω i ai + b i
+
#
−

+

∨ sup expN (ω)
n
ωi
i=k
(n)
ai
X

ω i (ai + bi ) = expN (ω)
i=k
ℓβ
X
i=k
− 
+ bi  
+ 
max(ℓα ,ℓβ )
ω i (ai + bi ) ∨ expN (ω)
+
ℓβ
X
ω i (ai + bi ) 
+
ω i (ai + bi ) ,
[α] ⋄ [β] =
+
+
= ⊤ → sup [α]+ ⋄ [β ′ + expN (ω)ω ℓβ −1 n]− ∨ sup [α′ + expN (ω)ω ℓα −1 n]− ⋄ [β]+

n
n
+
"
= ⊤ → sup expN (ω)
n

= ⊤ → expN (ω)
i=k
(n)
ω i ai + b i
max(ℓα ,ℓβ )
= ⊤ → expN (ω)

ℓα
X
X
i=k
max(ℓα ,ℓβ )
X
i=k
#
−
+
+

∨ sup expN (ω)
n


ω i (ai + bi ) = expN (ω)
(n)
ω i ai
i=k
ℓβ
X
i=k
X
i=k
− 
+ bi  
+ 
max(ℓα ,ℓβ )
ω i (ai + bi ) ∨ expN (ω)
+
ℓβ
X
ω i (ai + bi ) 
+
ω i (ai + bi ) .
158
Computations of Chapter 6
Next suppose that ℓα = 0, ℓβ > 0. Similarly,
[α]− ⋄ [β]− =
−
−
′
N
−
−
−
′
N
ℓβ −1 −
= ⊥ → sup [α] ⋄ [β + exp (ω)ω
n] ∨ sup [α + exp (n)] ⋄ [β]
n
n

"
"
#− 
#−
0
0
X
X
−
−
(n)
ω i ai + b i
ω i (a′i + bi ) + expN (n) 
= ⊥ → sup expN (ω)
∨ sup expN (ω)
n

= ⊥ → expN (ω)
"
n
i=k
N
= ⊥ → exp (ω)
−
ℓβ
X
ω i (ai + bi ) ∨ expN (ω)
i=k
0
X
i=k
"
#−
i
ω (ai + bi )
i=k
"
N
= exp (ω)
#− 
0
X
i=k
0
X

ω i (ai + bi )
#−
i
ω (ai + bi )
i=k
,
[α]− ⋄ [β]+ =
+
+
= ⊤ → sup [α]− ⋄ [β ′ + expN (ω)ω ℓβ −1 n]− ∨ sup [α′ + expN (n)]− ⋄ [β]+

n
n
+
"
= ⊤ → sup expN (ω)
n

= ⊤ → expN (ω)

= ⊤ → expN (ω)
+
−
0
X
ℓβ
X
i=k
ℓβ
X
i=k
i=k
(n)
ω i ai + b i
+
#−

+
∨ sup expN (ω)
n
ω i (ai + bi ) ∨ expN (ω)
+


ω i (ai + bi ) = expN (ω)
ℓβ
X
i=k
ℓβ
X
i=k
ℓβ
X
i=k
− 
ω i (a′i + bi ) 
+ 
ω i (a′i + bi ) 
+
ω i (ai + bi ) ,
[α] ⋄ [β] =
+
+
′
N
−
−
+
′
N
ℓβ −1 −
= ⊤ → sup [α] ⋄ [β + exp (ω)ω
n] ∨ sup [α + exp (n)] ⋄ [β]
n
n

#−
"
#− 
"
ℓ
0
α
X
X
+
+
(n)
ω i (a′i + bi ) + expN (n) 
ω i ai + b i
∨ sup expN (ω)
= ⊤ → sup expN (ω)
n

= ⊤ → exp (ω)
"
n
i=k
N
N
= ⊤ → exp (ω)
ℓβ
X
i=k
0
X
i=k
i
+
"
ω (ai + bi ) ∨ exp (ω)
i
#+
ω (ai + bi )
"
N
N
= exp (ω)
i=k
0
X
i
ω (ai + bi )
i=k
0
X
i=k
i
ω (ai + bi )
#+ 
#+

"
N
= exp (ω)
ℓα
X
i=k
i
#+
ω (ai + bi )
,
A.3. CLOSURE BY ⋄
159
[α]+ ⋄ [β]+ =
+
+
′
N
−
+
+
′
N
ℓβ −1 −
= ⊤ → sup [α] ⋄ [β + exp (ω)ω
n] ∨ sup [α + exp (n)] ⋄ [β]

n
n
+
"
= ⊤ → sup expN (ω)
n

= ⊤ → expN (ω)

= ⊤ → expN (ω)
ℓβ
X
i=k
ℓβ
X
i=k
ℓα
X
ω i ai +
i=k
(n)
bi
+
#−

+
∨ sup expN (ω)
n
ω i (ai + bi ) ∨ expN (ω)
+


ω i (ai + bi ) = expN (ω)
ℓβ
X
i=k
ℓβ
X
i=k
ℓβ
X
i=k
− 
ω i (a′i + bi ) 
+ 
ω i (a′i + bi ) 
+
ω i (ai + bi ) .
Finally, for ℓα = 0, ℓβ = 0
[α]− ⋄ [β]− =
−
−
′
N
−
−
−
′
N
−
= ⊥ → sup [α] ⋄ [β + exp (n)] ∨ sup [α + exp (n)] ⋄ [β]
n
n
= ⊥ → [θ]− ∨ [θ]− = ⊥ → [θ]− = [θ]− ,
[α]− ⋄ [β]+ =
+
+
′
N
−
+
−
′
N
−
= ⊤ → sup [α] ⋄ [β + exp (n)] ∨ sup [α + exp (n)] ⋄ [β]
n
n

"
#+ 
0
X
= ⊤ → [θ]+ ∨ expN (ω)
ω i (a′i + bi )  = ⊤ → [θ]+ = [θ]+ ,
i=k
[α]+ ⋄ [β]+ =
+
+
′
N
−
+
+
′
N
−
= ⊤ → sup [α] ⋄ [β + exp (n)] ∨ sup [α + exp (n)] ⋄ [β]
n
n
"
#+ "
#+ 
0
0
X
X
ω i (ai + b′i ) ∨ expN (ω)
= ⊤ →  expN (ω)
ω i (a′i + bi )  ,
i=k
and since
P0
i
′
i=k ω (ai + bi ) =
P0
i
′
i=k ω (ai + bi ) =
i=k
P
1
i=k
ω i (ai + bi ) + (a0 + b0 − 1), the claim follows.
Note that [0]µ is not a valid set. Nevertheless, to simplify notation we adopt a convention [α]µ ⋄ [0]ν = [α]µ .
PK
PL
Lemma A.7. Let [α]µ , [β]ν ∈ Φ and α =
i=N αi , β =
j=N βj , αi ∈ Φi , βj ∈ Φj , αK , βL > 0,
PJ+1
PJ+1
µ, ν ∈ {+, −}. For J < ω let αJ< = i=N αi , βJ< = i=N βi .
160
Computations of Chapter 6
• If K < L and [βL ]ν = [expL (ω)γ]−
[α]µ ⋄ [β]ν =
iµ
hP
K

→ [αL−1< + 1]− ⋄ [β]±
α

i
i=L−1


h
iµ

PK


α
−
1
→ [αL−1< + 1]± ⋄ [β]±
i=L−1 i


[1]+ → [αL−1< ]± ⋄ [β]ν




[αL + 1]− ⋄ [βL ]ν → [αL< + 1]− ⋄ [βL< + 1]−
for
PK
i=L−1
αi ≥ ω ,
PK
for ω > i=L−1 αi > 1 ,
iµ
hP
K
= [1]+ ,
for
α
i
hPi=L−1 iµ
K
for
= [1]− .
i=L−1 αi
• If K < L and [βL ]ν 6= [expL (ω)γ]−

−
ν

[αL−1< + 1] ⋄ [β]

+
±
ν
[α]µ ⋄ [β]ν = [1] → [αL−1< ] ⋄ [β]


[α + 1]− ⋄ [β ]ν → [α
L
L
L<
PK
α > 1,
hPi=L−1 i iµ
K
αi = [1]+ ,
for
hPi=L−1 iµ
K
for
= [1]− .
i=L−1 αi
for
+ 1]− ⋄ [βL< + 1]−
• If K = L

[αL ]µ ⋄ [βL ]ν → [αL< + 1]− ⋄ [βL< + 1]−



[α ]µ ⋄ [β − 1]ν → [α + 1]− ⋄ [β + 1]±
L
L
L<
L<
[α]µ ⋄ [β]ν =
µ
ν
±

[α
−
1]
⋄
[β
−
1]
→
[α
+
1]
⋄ [βL< + 1]±
L
L
L<



[αM + 1]µ ⋄ [βM + 1]ν → [αM< + 1]− ⋄ [βM< + 1]−
for
for
for
for
αL , βL ≥ ω ,
αL ≥ ω > βL > 1 ,
ω > αL · βL > 1 ,
αL · βL = 1,
where M is the least i > 1 such that αi or βi is nonzero.
Proof. Again, we prove all the equations simultaneously by induction on (α, β). It is easy to observe that the
inequality “≥” is in each case very simple. We concentrate on “≤”. Further more, since in all three cases the
arguments are very similar we only give the calculations for the first case: K < L and [βL ]ν = [expL (ω)γ]− .
′
Suppose first that αK = α′K + expK (ω)ω p , K > 0. Let α′ = αK< + α′K , β ′ = βK< + βK
. We only give the
calculation for p, q > 0 (if one or both are zero the calulation is entirely analogous):
µ
µ
[α]µ ⋄ [β]ν = 1µ → (sup[α′ + expK (ω)ω p−1 m]− ⋄ [β]− ) ∨ (sup [α]µ ⋄ [β ′ + expL (ω)ω q−1 n]+ )
m
n


" K+1
!
#−
X
µ
→ [αL−1< + 1]− ⋄ [β]±  ∨
= 1µ → sup
αi + α′K + expK (ω)ω p−1 m
m
i=L−1
µ
∨ (sup [αL−1< + 1]− ⋄ [β ′ + expL (ω)ω q−1 n]+ )
n
!
" K
#µ
X
µ
−
±
∨ ([αL−1< + 1]− ⋄ [β]µ )
≤1 →
αi → [αL−1< + 1] ⋄ [β]
=
"
K
X
i=L−1
i=L−1
#µ
αi
→ [αL−1< + 1]− ⋄ [β]± .
A.3. CLOSURE BY ⋄
161
Suppose now that αK = α′K + expK (p), K > 0. For q > 0
+
[α]µ ⋄ [β]ν = [expK (p)]µ → [α′ ]± ⋄ [β]− ∨ sup [α]µ ⋄ [β ′ + expL (ω)ω q−1 n]+ ) ∨
n
−
′
∨ sup [α] ⋄ [β + exp (ω)ω n]+ )
n
"

!
#±
K+1
X
≤ [expK (p)]µ → 
→ [αL−1< + 1]− ⋄ [β]±  ∨
αi + α′K
µ
L
q−1
i=L−1
−
∨ ([αL−1< + 1] ⋄ [β]+ ) ∨ ([αL−1< + 1]− ⋄ [β]− )
" K
#µ
X
≤
αi → [αL−1< + 1]− ⋄ [β]± ,
i=L−1
and for q = 0 the computation is analogous.
Finally suppose that K = 0. For αK = α′K + ω p and αK = α′K + 1 the calculations are entirely analogous
P
to the ones above save for three cases we consider below. For K
i=L−1 αi = ω
µ
µ
[α]µ ⋄ [β]ν = 1µ → (sup[α′ + m]− ⋄ [β]− ) ∨ (sup [α]µ ⋄ [βL′ + expL (ω)ω q−1 n]+ )
m
n
µ
≤ 1µ → sup [m − 1]− → [αL−1< + 1]± ⋄ [β]± ∨ ([αL−1< + 1]− ⋄ [β]µ )
m
µ
−
≤ 1µ → sup [m − 1] → 1+ → [αL−1< + 1]− ⋄ [β]± ∨ ([αL−1< + 1]− ⋄ [β]µ )
m
µ
≤ [ω] → [αL−1< + 1]− ⋄ [β]± ,
since [αL−1< + 1]+ ⋄ [β]± ≤ ⊤ → [αL−1< + 1]− ⋄ [β]− ≤ ⊤ → [αL−1< + 1]− ⋄ [β]± .
PK
For i=L−1 αi = 2
µ
[α]µ ⋄ [β]ν ≤ [1]µ → [α′ ]± ⋄ [β]− ∨ sup [α]µ ⋄ [β ′ + expL (ω)ω q−1 n]+ )
n
±
µ
≤ [1] → ([αL−1< + 1] ⋄ [β]µ ) ∨
µ
∨ sup [1]µ → [αL−1< + 1]± ⋄ [β ′ + expL (ω)ω q−1 n]±
n
µ
≤ [1]µ → ([αL−1< + 1]± ⋄ [β]µ ) ∨ sup [1]µ → [αL−1< + 1]± ⋄ [β]ν
n
±
≤ [1] → ([αL−1< + 1] ⋄ [β] ) ∨ [1]µ → [1]µ → [αL−1< + 1]± ⋄ [β]ν
µ
µ
= [1]µ → [αL−1< + 1]± ⋄ [β]± .
For
hP
K
i=L−1
αi
iµ
= [1]+
+
[α]µ ⋄ [β]ν = [1]+ → [αL−1< ]± ⋄ [β]− ∨ (sup [αL−1< + 1]+ ⋄ [β ′ + expL (ω)ω q−1 n]− )
±
−
n
+
±
−
n
+
±
−
≤ [1] → [αL−1< ] ⋄ [β] ∨ (sup [1]+ → [αL−1< ]± ⋄ [β ′ + expL (ω)ω q−1 n]− )
µ
≤ [1] → [αL−1< ] ⋄ [β] ∨ (sup [1]+ → [αL−1< ]± ⋄ [β]− )
µ
n
= [1] → [αL−1< ] ⋄ [β] ∨ ([1]+ → [1]+ → [αL−1< ]± ⋄ [β]− )
µ
= [1]µ → [αL−1< ]± ⋄ [β]−
162
For
hP
K
i=L−1
αi
Computations of Chapter 6
iµ
= [1]−
−
[α]µ ⋄ [β]ν = [1]− → [αL−1< ]± ⋄ [β]− ∨ (sup [αL−1< + 1]− ⋄ [β ′ + expL (ω)ω q−1 n]− )
n
≤ [1]− → [αL−1< ]± ⋄ [β]− ∨
−
∨ (sup [αL + 1]− ⋄ [βL′ + expL (ω)ω q−1 n]− → [αL< + 1]− ⋄ [βL< + 1]− )
n
≤ [1]− → [αL−1< ]± ⋄ [β]− ∨ ([αL + 1]− ⋄ [βL ]− → [αL< + 1]− ⋄ [βL< + 1]− )
If αL > 0 we get [αL−1< ]± ⋄ [β]− = [αL ]± ⋄ [βL ]− → [αL< + 1]− ⋄ [βL< + 1]− . If αL = 0, let N be
i−
hP
L
the least i > L such that αi > 0. Then we get [αL−1< ]− ⋄ [β]− = [αN −1< ]− ⋄ [β]− ≤
→
β
i
i=N −1
[αN −1< ]± ⋄ [βN −1< + 1]− = [βL ]− → [αN −1< ]± ⋄ [βL< ]− ≤ [βL ]− → [αL< + 1]− ⋄ [βL< + 1]− . In either case
we have
[α]µ ⋄ [β]ν ≤ [1]− → ([αL + 1]− ⋄ [βL ]− → [αL< + 1]− ⋄ [βL< + 1]− )
≤ [αL + 1]− ⋄ [βL ]− → [αL< + 1]− ⋄ [βL< + 1]− .
Case K < L and [βL ]ν = [βL′ + expL (ω)ω q ]+ . Suppose first that αK = α′K + expK (ω)ω p . Let α′ =
′
. We can treat K > 0 and K = 0 uniformly. We only give the calculation for
αK< + α′K , β ′ = βK< + βK
p, q > 0 (if one or both are zero its entirely analogous):
+
+
[α]µ ⋄ [β]ν = [1]+ → (sup[α′ + expK (ω)ω p−1 m]+ ⋄ [β]+ ) ∨ (sup [α]µ ⋄ [β ′ + expL (ω)ω q−1 n]+ )
m
n
+
+
= [1]+ → sup[αL−1< + 1]− ⋄ [β]+ ∨ (sup [αL−1< + 1]− ⋄ [β ′ + expL (ω)ω q−1 n]+ )
n
m
= [1]+ → [1]+ → [αL−1< + 1]− ⋄ [β]+ ∨ ([αL−1< + 1]− ⋄ [β]+ )
= [1]+ → [αL−1< + 1]− ⋄ [β]+ = [αL−1< + 1]− ⋄ [β]+ .
Suppose now that αK = α′K + expK (p) or αK = α′K + p with
+
PK
i=L−1
αi > 1. For q > 0
[α]µ ⋄ [β]ν = sup [α]µ ⋄ [β ′ + expL (ω)ω q−1 n]+
n
+
= sup [α′ + 1]− ⋄ [β ′ + expL (ω)ω q−1 n]+
n
= [αL−1< + 1]− ⋄ [β]+
and hfor q = 0 the
iµcomputation is analogous.
PK
For
= [1]+ we have
i=L−1 αi
+
[α]µ ⋄ [β]ν = sup [α]µ ⋄ [β ′ + expL (ω)ω q−1 n]+
n
+
= sup [1]+ → [αL−1< ]± ⋄ [β ′ + expL (ω)ω q−1 n]+
n
+
≤ sup [1]+ → [αL−1< ]± ⋄ [β]+
n
+
= [1] → [1]+ → [αL−1< ]± ⋄ [β]+
≤ [1]+ → [αL−1< ]± ⋄ [β]+ .
A.3. CLOSURE BY ⋄
For
hP
K
i=L−1
αi
iµ
163
= [1]−
+
[α]µ ⋄ [β]ν = sup [α]µ ⋄ [β ′ + expL (ω)ω q−1 n]+
n
+
= sup [αL + 1]− ⋄ [βL′ + expL (ω)ω q−1 n]+ → [αL< + 1]− ⋄ [βL< + 1]−
n
= [αL + 1]− ⋄ [βL ]+ → [αL< + 1]− ⋄ [βL< + 1]− .
Case K < L and [βL ]ν = [βL′ +expL (q)]ν . Suppose first that αK = α′K +expK (ω)ω p . Let α′ = αK< +α′K ,
′
β ′ = βK< + βK
. We can treat K > 0 and K = 0 uniformly. We only give the calculation for p > 0 (if p = 0
it is entirely analogous). For µ = − we have
−
+
[α]µ ⋄ [β]ν = [expL (q)]ν → [α]− ⋄ [β ′ ]± ∨ (sup[α′ + expK (ω)ω p−1 m]+ ⋄ [β]ν ) ∨ (sup [α′ + expK (ω)ω p−1 m]+ ⋄ [β]ν )
m
−
′ ±
+
ν
−
′ ±
m
+
−
n
−
= [exp (q)] → [αL−1< + 1] ⋄ [β ] ∨ (sup[αL−1< + 1] ⋄ [β] ) ∨ (sup [αL−1< + 1]− ⋄ [β]ν )
L
ν
n
= [exp (q)] → [αL−1< + 1] ⋄ [β ] ∨ ([1] → [αL−1< + 1] ⋄ [β] ) ∨ ([1]− → [αL−1< + 1]− ⋄ [β]ν )
L
ν
−
ν
= [expL (q)]ν → [1]± → [αL−1< + 1]− ⋄ [β]ν
= [αL−1< + 1]− ⋄ [β]ν ,
and for µ = +
+
[α]µ ⋄ [β]ν = sup[α′ + expK (ω)ω p−1 m]+ ⋄ [β]ν
m
+
= sup[αL−1< + 1]− ⋄ [β]ν
m
+
= [1] → [αL−1< + 1]− ⋄ [β]ν
= [αL−1< + 1]− ⋄ [β]ν .
Suppose now that αK = α′K + expK (p). We have
[α]µ ⋄ [β]ν = [expK (p)]µ ⋄ [expL (q)]ν → [α]µ ⋄ [β ′ ]± ∨ [α′ ]± ⋄ [β]ν
= [expL (q)]ν → [αL−1< + 1]− ⋄ [β ′ ]± ∨ [αL−1< + 1]− ⋄ [β]ν
= [expL (q)]ν → [αL−1< + 1]− ⋄ [β]ν
= [αL−1< + 1]− ⋄ [β]ν .
For αK = α′K + p with
have
PK
i=L−1 αi > 1 the computation is just like above. For
hP
K
[α]µ ⋄ [β]ν = [1]+ ⋄ [expL (q)]ν → [α]µ ⋄ [β ′ ]± ∨ [αL−1< ]± ⋄ [β]ν
= [1]+ → [αL−1< + 1]+ ⋄ [β ′ ]± ∨ [αL−1< ]± ⋄ [β]ν
If βL′ > 0, we can transform this further as
= [1]+ → (([1]+ → [αL−1< ]± ⋄ [β ′ ]± ) ∨ [αL−1< ]± ⋄ [β]ν )
= [1]+ → (([αL−1< ]± ⋄ [β ′ ]± ) ∨ [αL−1< ]± ⋄ [β]ν )
= [1]+ → ([αL−1< ]± ⋄ [β]ν )
i=L−1 αi
iµ
= [1]+ we
164
Computations of Chapter 6
′
and we are done. Suppose βL′ = 0. Let M be the least i > L such that βi > 0. If βM = βM
+ expM (ω)ω r ,
then we have
" L
#+
1
X
+
= [1] → ((
αi + 1
→ [αM−1< ]± ⋄ [β ′ ]± ) ∨ [αL−1< ]± ⋄ [β]ν )
i=M−1
+
= [1] → (([1] → [αL−1< ]± ⋄ [β ′ ]± ) ∨ [αL−1< ]± ⋄ [β]ν )
+
= [1]+ → (([αL−1< ]± ⋄ [β ′ ]± ) ∨ [αL−1< ]± ⋄ [β]ν )
= [1]+ → ([αL−1< ]± ⋄ [β]ν ) .
′
If βM = βM
+ expM (r), then we have
= [1]+ → (([αM−1< ]± ⋄ [β ′ ]± ) ∨ [αL−1< ]± ⋄ [β]ν )
= [1]+ → (([αL−1< ]± ⋄ [β ′ ]± ) ∨ [αL−1< ]± ⋄ [β]ν )
= [1]+ → ([αL−1< ]± ⋄ [β]ν ) .
For
hP
K
i=L−1
αi
iµ
= [1]−
[α]µ ⋄ [β]ν = [1]− ⋄ [expL (q)]ν → [α]µ ⋄ [β ′ ]± ∨ [αL−1< ]± ⋄ [β]ν
= [expL (q)]ν → [αL−1< + 1]− ⋄ [β ′ ]± ∨ [αL−1< ]± ⋄ [β]ν
= [βL ]ν → [αL< + 1]− ⋄ [βL< + 1]−
= [αL + 1]− ⋄ [βL ]ν → [αL< + 1]− ⋄ [βL< + 1]−
Case K = L. Using similar computations as above, follow the subcases of Lemma A.6.
Lemma A.7 gives a recursive procedure to compute the result of [α]µ ⋄ [β]ν . Observe that in each case we
reduce the problem to at most two instances: one on elements of (roughly speaking) the same ΦN , the other
on sets corresponding to ordinals such that either their respective K and L are smaller, or are obtained by
cutting off a nontrivial (bigger than 1) tail part of the sum defining the ordinal, and replacing it with 1.
The first subproblem is solved by Lemma A.6, and the second problem solved recursively.
What remains to be done, is put the results of the two
PL+1together. For this we need to prove
Psubcomputations
0
that the class Φ is closed by operations of the form [ i=L αi ]µ → [ i=M αi + j]ν with j < ω, and αi ∈ Φi .
P
P
P
P
ν
±
Clearly, [ 0i=L αi ]µ → [ L+1
[ 0i=L αi ]µ → [ L+1
i=M αi + j] =P
i=M αi ] , if j > 0.
P0
P
L+1
0
µ
ν
µ
If ν = ±, [ i=L αi ] → [ i=M αi ] = [ i=M αi ] . If ν = + or ν = −, the result depends on the form of
αL (w.l.o.g we may suppose that it is nonzero).
P0
PL
If αL = α′L + expL (ω)ω p , then the result is [ i=M αi ]µ as well. Otherwise, it is [ i=M αi ]ν .
This concludes the proof of the effective closure by ⋄.
A.4
Closure by sup+ and sup−
µ hii
We show that for [α]µ , [β]ν ∈ Φ, we can compute sup+
⋄ [β]ν ) and that the result is in Φ. The case
i (([α] )
−
µ hii
ν
supi (([α] ) ⋄ [β] ) is solved in the same way.
µ hii
Case α = expN (ω)ω p + α′ . If [α]µ = [expN (ω)ω p ]+ , then ([α]µ )hii = [α]µ and sup+
⋄ [β]ν ) =
i (([α] )
+
µ
ν
+
µ
ν
supi ([α] ⋄ [β] ) = [1] → [α] ⋄ [β] . We conclude by previously proved closure properties.
Otherwise, [α]µ ≥ [expN (ω)ω p ]− . But then, by the lemmas of the previous section, it follows easily that
([α]µ )hii ≤ ([expN (ω)ω p ]− )hi+1i ≤ [expN (ω)ω p (i + 2)]+ . Thus,
+
+
+
i
i
i
sup(([α]µ )hii ⋄ [β]ν ) = sup([expN (ω)ω p i]− ⋄ [β]ν ) = sup([expN (ω)ω p i]+ ⋄ [β]ν ).
A.4. CLOSURE BY SUP+ AND SUP−
165
PL
Let n=M βn be the usual presentation of β with βL > 0. There are only three cases to consider, all the
others being verified just by combining those three.
µ hii
(1) Let L > N . If ν = +, then [expN (ω)ω p i]+ ⋄ [β]ν = [β]ν and sup+
⋄ [β]ν ) = [1]+ → [β]ν . We
i (([α] )
conclude by previously proved closure property for →.
Pk
Pk
For ν 6= +, let βL = expL (ω)η + j=1 expL (j)kj , with η < ω ω and kj < ω. If j=1 expL (j)kj = 0,
then by the properties of ⋄ we obtain that [expN (ω)ω p i]+ ⋄ [β]ν = [β + expN (ω)ω p i]+ and therefore
+
+
i
i
sup(([α]µ )hii ⋄ [β]ν ) = sup([β + expN (ω)ω p i]+ ) = [β + expN (ω)ω p+1 ]+ .
µ hii
Otherwise, since [expN (ω)ω p i]+ ⋄[expL (p)]ν = [expL (p)]ν , [expN (ω)ω p i]+ ⋄[β]ν = [β]ν and sup+
i (([α] ) ⋄
ν
+
ν
[β] ) = [1] → [β] . As before, we conclude by previously proved closure properties.
µ hii
(2) If M < N , [expN (ω)ω p i]+ ⋄ [β]ν = [expN (ω)ω p i]+ , and sup+
⋄ [β]ν ) = [expN (ω)ω p+1 ]+ .
i (([α] )
′
′
(3) Suppose that L = N . Let βN = expN (ω)(ω p+1 η+ω p j)+βN
with η < ω ω , j < ω, and βN
< expN (ω)ω p .
Then, by applying the properties of the operation ⋄ given by Lemma A.6, we reason as follows.
PN +1
′
If ν = +, βN
= 0, and j = 0, we get [expN (ω)ω p i]+ ⋄ [β]ν = [ n=M βn + expN (ω)ηω p ]+ , and since i
does not occur in the previous formula, we obtain that
+
sup(([α]µ )hii ⋄ [β]+ ) = [
i
N
+1
X
βn + expN (ω)ω p+1 η]+ .
n=M
′
Suppose ν 6= + or βN
> 0. Under each of the two conditions we have
[expN (ω)ω p i]+ ⋄ [β]ν = [
N
+1
X
βn + expN (ω)(ω p+1 η + ω p (i + j))]+
n=M
and so
+
sup(([α]µ )hii ⋄ [β]ν ) = [
i
N
+1
X
βn + expN (ω)ω p+1 (η + 1)]+ .
n=M
′
Finally, if ν = +, βN
= 0, and j > 0, it holds that
[expN (ω)ω p i]+ ⋄ [β]ν = [
N
+1
X
βn + expN (ω)(ω p+1 η + ω p (i + j − 1))]+
n=M
and we conclude like before.
Case α = expN (p) + α′ or α < ω. The case α < ω is solved by similar arguments as the previous case.
So, let α = expN (p) + α′ . If α = expN (1) and µ ∈ {+, −}, we get ([α]µ )hii = [α]µ , and the claim follows.
Otherwise, [α] ≥W [expN (1)]± . Like before, by the properties of ⋄, we have
+
+
i
i
+
sup(([α]µ )hii ⋄ [β]ν ) = sup(([expN (p)]µ )hii ⋄ [β]ν ) = sup([expN (i)]+ ⋄ [β]ν ).
i
PL
µ hii
⋄ [β]ν ) = [1]+ . Let
If [β]ν = [1]+ , then sup+
i (([α] )
n=M βn be the usual presentation of β with
ν
−
[βL ] ≥W [1] . Again, there are only three cases to consider, all the others being verified just by combining
those three.
µ hii
(1) If M < N , [expN (i)]+ ⋄ [β]ν = [expN (i)]+ , and sup+
⋄ [β]ν ) = [expN (ω)]+ .
i (([α] )
166
Computations of Chapter 6
µ hii
(2) Let L > N . If ν = +, then [expN (i)]+ ⋄ [β]ν = [β]ν and sup+
⋄ [β]ν ) = [1]+ → [β]ν . We
i (([α] )
conclude by previously proved closure property for →.
P
P
For ν 6= +, let βL = expL (ω)η + kj=1 expL (j)kj , with η < ω ω and kj < ω. If kj=1 expL (j)kj = 0,
then [expN (i)]+ ⋄ [β]ν = [β + expN (i)]+ and therefore
+
+
i
i
sup(([α]µ )hii ⋄ [β]ν ) = sup([β + expN (i)]+ ) = [β + expN (ω)]+ .
µ hii
Otherwise, since [expN (i)]+ ⋄ [expL (p)]ν = [expL (p)]ν , [expN (i)]+ ⋄ [β]ν = [β]ν and sup+
⋄
i (([α] )
ν
+
ν
[β] ) = [1] → [β] . As before, we conclude by previously proved closure properties.
P
(3) Suppose that L = N . Let βN = expN (ω)η + kj=1 expN (j)kj with η < ω ω , kj < ω. Then, by applying
the properties of the operation ⋄, we reason as follows.
Pk
µ hii
If ν = + and j=1 expN (j)kj = 0, we get [expN (i)]+ ⋄ [β]ν = [β]ν , and we obtain that sup+
⋄
i (([α] )
ν
+
ν
[β] ) = [1] → [β] .
Pk
Suppose ν 6= + or j=1 expN (j)kj > 0. Under each of the two conditions we have
[expN (i)]+ ⋄ [β]ν ≤W [expN (ω)η + expN (p)]+ ,
for a certain 0 < p < ω and so
+
sup(([α]µ )hii ⋄ [β]ν ) = [expN (ω)η + expN (ω)]+ = [expN (ω)(η + 1)]+ .
i
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